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5.2 - writing hypotheses.

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)).

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

• At this point we can write hypotheses for a single mean (\(\mu\)), paired means(\(\mu_d\)), a single proportion (\(p\)), the difference between two independent means (\(\mu_1-\mu_2\)), the difference between two proportions (\(p_1-p_2\)), a simple linear regression slope (\(\beta\)), and a correlation (\(\rho\)). 
• The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
• The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., \(\mu_0\) and \(p_0\)). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., \(p\) and \(\mu\)).  The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

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• How to Write a Strong Hypothesis | Guide & Examples

How to Write a Strong Hypothesis | Guide & Examples

Published on 6 May 2022 by Shona McCombes .

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more variables . An independent variable is something the researcher changes or controls. A dependent variable is something the researcher observes and measures.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1: ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2: Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalise more complex constructs.

Step 3: Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

Step 4: Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

Step 5: Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in if … then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

Step 6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis. The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis is not just a guess. It should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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How to Write a Research Hypothesis

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Since grade school, we've all been familiar with hypotheses. The hypothesis is an essential step of the scientific method. But what makes an effective research hypothesis, how do you create one, and what types of hypotheses are there? We answer these questions and more.

Updated on April 27, 2022

What is a research hypothesis?

General hypothesis.

Since grade school, we've all been familiar with the term “hypothesis.” A hypothesis is a fact-based guess or prediction that has not been proven. It is an essential step of the scientific method. The hypothesis of a study is a drive for experimentation to either prove the hypothesis or dispute it.

Research Hypothesis

A research hypothesis is more specific than a general hypothesis. It is an educated, expected prediction of the outcome of a study that is testable.

What makes an effective research hypothesis?

A good research hypothesis is a clear statement of the relationship between a dependent variable(s) and independent variable(s) relevant to the study that can be disproven.

Research hypothesis checklist

Once you've written a possible hypothesis, make sure it checks the following boxes:

• It must be testable: You need a means to prove your hypothesis. If you can't test it, it's not a hypothesis.
• It must include a dependent and independent variable: At least one independent variable ( cause ) and one dependent variable ( effect ) must be included.
• The language must be easy to understand: Be as clear and concise as possible. Nothing should be left to interpretation.
• It must be relevant to your research topic: You probably shouldn't be talking about cats and dogs if your research topic is outer space. Stay relevant to your topic.

How to create an effective research hypothesis

Pose it as a question first.

Start your research hypothesis from a journalistic approach. Ask one of the five W's: Who, what, when, where, or why.

A possible initial question could be: Why is the sky blue?

Do the preliminary research

Once you have a question in mind, read research around your topic. Collect research from academic journals.

If you're looking for information about the sky and why it is blue, research information about the atmosphere, weather, space, the sun, etc.

Write a draft hypothesis

Once you're comfortable with your subject and have preliminary knowledge, create a working hypothesis. Don't stress much over this. Your first hypothesis is not permanent. Look at it as a draft.

Your first draft of a hypothesis could be: Certain molecules in the Earth's atmosphere are responsive to the sky being the color blue.

Make your working draft perfect

Take your working hypothesis and make it perfect. Narrow it down to include only the information listed in the “Research hypothesis checklist” above.

Now that you've written your working hypothesis, narrow it down. Your new hypothesis could be: Light from the sun hitting oxygen molecules in the sky makes the color of the sky appear blue.

Write a null hypothesis

Your null hypothesis should be the opposite of your research hypothesis. It should be able to be disproven by your research.

In this example, your null hypothesis would be: Light from the sun hitting oxygen molecules in the sky does not make the color of the sky appear blue.

Why is it important to have a clear, testable hypothesis?

One of the main reasons a manuscript can be rejected from a journal is because of a weak hypothesis. “Poor hypothesis, study design, methodology, and improper use of statistics are other reasons for rejection of a manuscript,” says Dr. Ish Kumar Dhammi and Dr. Rehan-Ul-Haq in Indian Journal of Orthopaedics.

According to Dr. James M. Provenzale in American Journal of Roentgenology , “The clear declaration of a research question (or hypothesis) in the Introduction is critical for reviewers to understand the intent of the research study. It is best to clearly state the study goal in plain language (for example, “We set out to determine whether condition x produces condition y.”) An insufficient problem statement is one of the more common reasons for manuscript rejection.”

Characteristics that make a hypothesis weak include:

• Unclear variables
• Unoriginality
• Too general
• Too specific

A weak hypothesis leads to weak research and methods . The goal of a paper is to prove or disprove a hypothesis - or to prove or disprove a null hypothesis. If the hypothesis is not a dependent variable of what is being studied, the paper's methods should come into question.

A strong hypothesis is essential to the scientific method. A hypothesis states an assumed relationship between at least two variables and the experiment then proves or disproves that relationship with statistical significance. Without a proven and reproducible relationship, the paper feeds into the reproducibility crisis. Learn more about writing for reproducibility .

In a study published in The Journal of Obstetrics and Gynecology of India by Dr. Suvarna Satish Khadilkar, she reviewed 400 rejected manuscripts to see why they were rejected. Her studies revealed that poor methodology was a top reason for the submission having a final disposition of rejection.

Aside from publication chances, Dr. Gareth Dyke believes a clear hypothesis helps efficiency.

“Developing a clear and testable hypothesis for your research project means that you will not waste time, energy, and money with your work,” said Dyke. “Refining a hypothesis that is both meaningful, interesting, attainable, and testable is the goal of all effective research.”

Types of research hypotheses

There can be overlap in these types of hypotheses.

Simple hypothesis

A simple hypothesis is a hypothesis at its most basic form. It shows the relationship of one independent and one independent variable.

Example: Drinking soda (independent variable) every day leads to obesity (dependent variable).

Complex hypothesis

A complex hypothesis shows the relationship of two or more independent and dependent variables.

Example: Drinking soda (independent variable) every day leads to obesity (dependent variable) and heart disease (dependent variable).

Directional hypothesis

A directional hypothesis guesses which way the results of an experiment will go. It uses words like increase, decrease, higher, lower, positive, negative, more, or less. It is also frequently used in statistics.

Example: Humans exposed to radiation have a higher risk of cancer than humans not exposed to radiation.

Non-directional hypothesis

A non-directional hypothesis says there will be an effect on the dependent variable, but it does not say which direction.

Associative hypothesis

An associative hypothesis says that when one variable changes, so does the other variable.

Alternative hypothesis

An alternative hypothesis states that the variables have a relationship.

• The opposite of a null hypothesis

Example: An apple a day keeps the doctor away.

Null hypothesis

A null hypothesis states that there is no relationship between the two variables. It is posed as the opposite of what the alternative hypothesis states.

Researchers use a null hypothesis to work to be able to reject it. A null hypothesis:

• Can never be proven
• Can only be rejected
• Is the opposite of an alternative hypothesis

Example: An apple a day does not keep the doctor away.

Logical hypothesis

A logical hypothesis is a suggested explanation while using limited evidence.

Example: Bats can navigate in the dark better than tigers.

In this hypothesis, the researcher knows that tigers cannot see in the dark, and bats mostly live in darkness.

Empirical hypothesis

An empirical hypothesis is also called a “working hypothesis.” It uses the trial and error method and changes around the independent variables.

• An apple a day keeps the doctor away.
• Two apples a day keep the doctor away.
• Three apples a day keep the doctor away.

In this case, the research changes the hypothesis as the researcher learns more about his/her research.

Statistical hypothesis

A statistical hypothesis is a look of a part of a population or statistical model. This type of hypothesis is especially useful if you are making a statement about a large population. Instead of having to test the entire population of Illinois, you could just use a smaller sample of people who live there.

Example: 70% of people who live in Illinois are iron deficient.

Causal hypothesis

A causal hypothesis states that the independent variable will have an effect on the dependent variable.

Example: Using tobacco products causes cancer.

Final thoughts

Make sure your research is error-free before you send it to your preferred journal . Check our our English Editing services to avoid your chances of desk rejection.

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Module: Inference for One Proportion

Hypothesis testing (1 of 5), given a claim about a population, construct an appropriate set of hypotheses to test and properly interpret p values and type i / ii errors., learning objectives.

• When testing a claim, distinguish among situations involving one population mean, one population proportion, two population means, or two population proportions.
• Given a claim about a population, determine null and alternative hypotheses.

Introduction

In inference, we use a sample to draw a conclusion about a population. Two types of inference are the focus of our work in this course:

• Estimate a population parameter with a confidence interval.
• Test a claim about a population parameter with a hypothesis test.

We can also use samples from two populations to compare those populations. In this situation, the two types of inference focus on differences in the parameters.

• Estimate a difference in population parameters with a confidence interval.
• Test a claim about a difference in population parameters with a hypothesis test.

In “Estimating a Population Proportion,” we learned to estimate a population proportion using a confidence interval. For example, we estimated the proportion of all Tallahassee Community College students who are female and the proportion of all American adults who used the Internet to obtain medical information in the previous month. We will revisit confidence intervals in future modules.

Now we look more carefully at how to test a claim with a hypothesis test. Statistical investigations begin with research questions. We begin our discussion of hypothesis tests with research questions that require us to test a claim. Later we look at how a claim becomes a hypothesis.

Research Questions about Testing Claims

Let’s revisit some of the research questions from examples in the module Types of Statistical Studies and Producing Data that involve testing a claim.

Is the average course load for community college students less than 12 semester hours? This question contains a claim about a population mean. The question contains information about the population, the variable, and the parameter. The population is all community college students. The variable is course load in semester hours . It is quantitative, so the parameter is a mean. The claim is, “The mean course load for all community college students is less than 12 semester hours.”

Do the majority of community college students qualify for federal student loans? This question contains a claim about a population proportion and information about the population, the variable, and the parameter. The population is all community college students. The variable is Qualify for federal student loan (yes or no). It is categorical, so the parameter is a proportion. The claim is, “The proportion of community college students who qualify is greater than 0.5” (a majority means more than half, or 0.5).

In community colleges, do female students and male students have different mean GPAs? This question contains a claim that compares two population means. Again, we see information about the populations, the variable, and the parameters. The two populations are female community college students and male community college students. The variable is GPA . It is quantitative, so the parameters are means. The claim is, “The mean GPA for female community college students is different from the mean GPA for male community college students.” Notice that the claim compares the two population means, but there is no claim about the numeric value of either mean.

Are college athletes more likely than nonathletes to receive academic advising? This question contains a claim that compares two population proportions: college athletes and college students who are not athletes. The variable is Receive academic advising (yes or no). The variable is categorical, so the parameters are proportions. The claim is, “The proportion of all college athletes who receive academic advising is greater than the proportion of all nonathletes in college who receive academic advising.” Notice that the claim compares two population proportions, but there is no claim about the numeric value of either proportion.

In the case of testing a claim about a single population parameter, we compare it to a numeric value. In the case of testing a claim about two population parameters, we compare them to each other.

Learn By Doing

Identify the type of claim in each research question below.

Next Steps: Forming Hypotheses

We already know that in inference we use a sample to draw a conclusion about a population. If the research question contains a claim about the population, we translate the claim into two related hypotheses.

The null hypothesis is a hypothesis about the value of the parameter. The null hypothesis relates to our work in Linking Probability to Statistical Inference where we drew a conclusion about a population parameter on the basis of the sampling distribution. We started with an assumption about the value of the parameter, then used a simulation to simulate the selection of random samples from a population with this parameter value. Or we used the parameter value in a mathematical model to describe the center and spread of the sampling distribution. The null hypothesis gives the value of the parameter that we will use to create the sampling distribution. In this way, the null hypothesis states what we assume to be true about the population.

The alternative hypothesis usually reflects the claim in the research question about the value of the parameter. The alternative hypothesis says the parameter is “greater than” or “less than” or “not equal to” the value we assume to true in the null hypothesis.

Stating Hypotheses

Here are the hypotheses for the research questions from the previous example. The null hypothesis is abbreviated H 0 . The alternative hypothesis is abbreviated H a .

Is the average course load for community college students less than 12 semester hours?

• H 0 : The mean course load for community college students is equal to 12 semester hours.
• H a : The mean course load for community college students is less than 12 semester hours.

Do the majority of community college students qualify for federal student loans?

• H 0 : The proportion of community college students who qualify for federal student loans is 0.5.
• H a : The proportion of community college students who qualify for federal student loans is greater than 0.5.

When the research question contains a claim that compares two populations, the null hypothesis states that the parameters are equal. We will see in Modules 9 and 10 that we translate the null hypothesis into a statement about “no difference” in parameter values. We revisit this idea in more depth later.

In community colleges, do female students and male students have different mean GPAs?

• H 0 : In community colleges, female and male students have the same mean GPAs.
• H a : In community colleges, female and male students have different mean GPAs.

Are college athletes more likely than nonathletes to receive academic advising?

• H 0 : In colleges, the proportion of athletes who receive academic advising is equal to the proportion of nonathletes who receive academic advising.
• H a : In colleges, the proportion of athletes who receive academic advising is greater than the proportion of nonathletes who receive academic advising.

Here are some general observations about null and alternative hypotheses.

• The hypotheses are competing claims about the parameter or about the comparison of parameters.
• Both hypotheses are statements about the same population parameter or same two population parameters.
• The null hypothesis contains an equal sign.
• The alternative hypothesis is always an inequality statement. It contains a “less than” or a “greater than” or a “not equal to” symbol.
• In a statistical investigation, we determine the research question, and thus the hypotheses, before we collect data.

The process of forming hypotheses, collecting data, and using the data to draw a conclusion about the hypotheses is called hypothesis testing .

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Statistics Made Easy

How to Write a Null Hypothesis (5 Examples)

A hypothesis test uses sample data to determine whether or not some claim about a population parameter is true.

Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms:

H 0 (Null Hypothesis): Population parameter =,  ≤, ≥ some value

H A  (Alternative Hypothesis): Population parameter <, >, ≠ some value

Note that the null hypothesis always contains the equal sign .

We interpret the hypotheses as follows:

Null hypothesis: The sample data provides no evidence to support some claim being made by an individual.

Alternative hypothesis: The sample data  does provide sufficient evidence to support the claim being made by an individual.

For example, suppose it’s assumed that the average height of a certain species of plant is 20 inches tall. However, one botanist claims the true average height is greater than 20 inches.

To test this claim, she may go out and collect a random sample of plants. She can then use this sample data to perform a hypothesis test using the following two hypotheses:

H 0 : μ ≤ 20 (the true mean height of plants is equal to or even less than 20 inches)

H A : μ > 20 (the true mean height of plants is greater than 20 inches)

If the sample data gathered by the botanist shows that the mean height of this species of plants is significantly greater than 20 inches, she can reject the null hypothesis and conclude that the mean height is greater than 20 inches.

Read through the following examples to gain a better understanding of how to write a null hypothesis in different situations.

Example 1: Weight of Turtles

A biologist wants to test whether or not the true mean weight of a certain species of turtles is 300 pounds. To test this, he goes out and measures the weight of a random sample of 40 turtles.

Here is how to write the null and alternative hypotheses for this scenario:

H 0 : μ = 300 (the true mean weight is equal to 300 pounds)

H A : μ ≠ 300 (the true mean weight is not equal to 300 pounds)

Example 2: Height of Males

It’s assumed that the mean height of males in a certain city is 68 inches. However, an independent researcher believes the true mean height is greater than 68 inches. To test this, he goes out and collects the height of 50 males in the city.

H 0 : μ ≤ 68 (the true mean height is equal to or even less than 68 inches)

H A : μ > 68 (the true mean height is greater than 68 inches)

Example 3: Graduation Rates

A university states that 80% of all students graduate on time. However, an independent researcher believes that less than 80% of all students graduate on time. To test this, she collects data on the proportion of students who graduated on time last year at the university.

H 0 : p ≥ 0.80 (the true proportion of students who graduate on time is 80% or higher)

H A : μ < 0.80 (the true proportion of students who graduate on time is less than 80%)

Example 4: Burger Weights

A food researcher wants to test whether or not the true mean weight of a burger at a certain restaurant is 7 ounces. To test this, he goes out and measures the weight of a random sample of 20 burgers from this restaurant.

H 0 : μ = 7 (the true mean weight is equal to 7 ounces)

H A : μ ≠ 7 (the true mean weight is not equal to 7 ounces)

Example 5: Citizen Support

A politician claims that less than 30% of citizens in a certain town support a certain law. To test this, he goes out and surveys 200 citizens on whether or not they support the law.

H 0 : p ≥ .30 (the true proportion of citizens who support the law is greater than or equal to 30%)

H A : μ < 0.30 (the true proportion of citizens who support the law is less than 30%)

Additional Resources

Introduction to Hypothesis Testing Introduction to Confidence Intervals An Explanation of P-Values and Statistical Significance

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17 Introduction to Hypothesis Testing

Jenna Lehmann

What is Hypothesis Testing?

Hypothesis testing is a big part of what we would actually consider testing for inferential statistics. It’s a procedure and set of rules that allow us to move from descriptive statistics to make inferences about a population based on sample data. It is a statistical method that uses sample data to evaluate a hypothesis about a population.

This type of test is usually used within the context of research. If we expect to see a difference between a treated and untreated group (in some cases the untreated group is the parameters we know about the population), we expect there to be a difference in the means between the two groups, but that the standard deviation remains the same, as if each individual score has had a value added or subtracted from it.

Steps of Hypothesis Testing

The following steps will be tailored to fit the first kind of hypothesis testing we will learn first: single-sample z-tests. There are many other kinds of tests, so keep this in mind.

• Null Hypothesis (H0): states that in the general population there is no change, no difference, or no relationship, or in the context of an experiment, it predicts that the independent variable has no effect on the dependent variable.
• Alternative Hypothesis (H1): states that there is a change, a difference, or a relationship for the general population, or in the context of an experiment, it predicts that the independent variable has an effect on the dependent variable.

• Critical Region: Composed of the extreme sample values that are very unlikely to be obtained if the null hypothesis is true. Determined by alpha level. If sample data fall in the critical region, the null hypothesis is rejected, because it’s very unlikely they’ve fallen there by chance.
• After collecting the data, we find the sample mean. Now we can compare the sample mean with the null hypothesis by computing a z-score that describes where the sample mean is located relative to the hypothesized population mean. We use the z-score formula.
• We decided previously what the two z-score boundaries are for a critical score. If the z-score we get after plugging the numbers in the aforementioned equation is outside of that critical region, we reject the null hypothesis. Otherwise, we would say that we failed to reject the null hypothesis.

Regions of the Distribution

Because we’re making judgments based on probability and proportion, our normal distributions and certain regions within them come into play.

The Critical Region is composed of the extreme sample values that are very unlikely to be obtained if the null hypothesis is true. Determined by alpha level. If sample data fall in the critical region, the null hypothesis is rejected, because it’s very unlikely they’ve fallen there by chance.

These regions come into play when talking about different errors.

A Type I Error occurs when a researcher rejects a null hypothesis that is actually true; the researcher concludes that a treatment has an effect when it actually doesn’t. This happens when a researcher unknowingly obtains an extreme, non-representative sample. This goes back to alpha level: it’s the probability that the test will lead to a Type I error if the null hypothesis is true.

A result is said to be significant or statistically significant if it is very unlikely to occur when the null hypothesis is true. That is, the result is sufficient to reject the null hypothesis. For instance, two means can be significantly different from one another.

Factors that Influence and Assumptions of Hypothesis Testing

Assumptions of Hypothesis Testing:

• Random sampling: it is assumed that the participants used in the study were selected randomly so that we can confidently generalize our findings from the sample to the population.
• Independent observation: two observations are independent if there is no consistent, predictable relationship between the first observation and the second. The value of σ is unchanged by the treatment; if the population standard deviation is unknown, we assume that the standard deviation for the unknown population (after treatment) is the same as it was for the population before treatment. There are ways of checking to see if this is true in SPSS or Excel.
• Normal sampling distribution: in order to use the unit normal table to identify the critical region, we need the distribution of sample means to be normal (which means we need the population to be distributed normally and/or each sample size needs to be 30 or greater based on what we know about the central limit theorem).

Factors that influence hypothesis testing:

• The variability of the scores, which is measured by either the standard deviation or the variance. The variability influences the size of the standard error in the denominator of the z-score.
• The number of scores in the sample. This value also influences the size of the standard error in the denominator.

Test statistic: indicates that the sample data are converted into a single, specific statistic that is used to test the hypothesis (in this case, the z-score statistic).

Directional Hypotheses and Tailed Tests

In a directional hypothesis test , also known as a one-tailed test, the statistical hypotheses specify with an increase or decrease in the population mean. That is, they make a statement about the direction of the effect.

The Hypotheses for a Directional Test:

• H0: The test scores are not increased/decreased (the treatment doesn’t work)
• H1: The test scores are increased/decreased (the treatment works as predicted)

Because we’re only worried about scores that are either greater or less than the scores predicted by the null hypothesis, we only worry about what’s going on in one tail meaning that the critical region only exists within one tail. This means that all of the alpha is contained in one tail rather than split up into both (so the whole 5% is located in the tail we care about, rather than 2.5% in each tail). So before, we cared about what’s going on at the 0.025 mark of the unit normal table to look at both tails, but now we care about 0.05 because we’re only looking at one tail.

A one-tailed test allows you to reject the null hypothesis when the difference between the sample and the population is relatively small, as long as that difference is in the direction that you predicted. A two-tailed test, on the other hand, requires a relatively large difference independent of direction. In practice, researchers hypothesize using a one-tailed method but base their findings off of whether the results fall into the critical region of a two-tailed method. For the purposes of this class, make sure to calculate your results using the test that is specified in the problem.

Effect Size

A measure of effect size is intended to provide a measurement of the absolute magnitude of a treatment effect, independent of the size of the sample(s) being used. Usually done with Cohen’s d. If you imagine the two distributions, they’re layered over one another. The more they overlap, the smaller the effect size (the means of the two distributions are close). The more they are spread apart, the greater the effect size (the means of the two distributions are farther apart).

Statistical Power

The power of a statistical test is the probability that the test will correctly reject a false null hypothesis. It’s usually what we’re hoping to get when we run an experiment. It’s displayed in the table posted above. Power and effect size are connected. So, we know that the greater the distance between the means, the greater the effect size. If the two distributions overlapped very little, there would be a greater chance of selecting a sample that leads to rejecting the null hypothesis.

This chapter was originally posted to the Math Support Center blog at the University of Baltimore on June 11, 2019.

Math and Statistics Guides from UB's Math & Statistics Center Copyright © by Jenna Lehmann is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Hypothesis Testing

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CO-6: Apply basic concepts of probability, random variation, and commonly used statistical probability distributions.

Learning Objectives

LO 6.26: Outline the logic and process of hypothesis testing.

LO 6.27: Explain what the p-value is and how it is used to draw conclusions.

Video: Hypothesis Testing (8:43)

Introduction

We are in the middle of the part of the course that has to do with inference for one variable.

So far, we talked about point estimation and learned how interval estimation enhances it by quantifying the magnitude of the estimation error (with a certain level of confidence) in the form of the margin of error. The result is the confidence interval — an interval that, with a certain confidence, we believe captures the unknown parameter.

We are now moving to the other kind of inference, hypothesis testing . We say that hypothesis testing is “the other kind” because, unlike the inferential methods we presented so far, where the goal was estimating the unknown parameter, the idea, logic and goal of hypothesis testing are quite different.

In the first two parts of this section we will discuss the idea behind hypothesis testing, explain how it works, and introduce new terminology that emerges in this form of inference. The final two parts will be more specific and will discuss hypothesis testing for the population proportion ( p ) and the population mean ( μ, mu).

If this is your first statistics course, you will need to spend considerable time on this topic as there are many new ideas. Many students find this process and its logic difficult to understand in the beginning.

In this section, we will use the hypothesis test for a population proportion to motivate our understanding of the process. We will conduct these tests manually. For all future hypothesis test procedures, including problems involving means, we will use software to obtain the results and focus on interpreting them in the context of our scenario.

General Idea and Logic of Hypothesis Testing

The purpose of this section is to gradually build your understanding about how statistical hypothesis testing works. We start by explaining the general logic behind the process of hypothesis testing. Once we are confident that you understand this logic, we will add some more details and terminology.

To start our discussion about the idea behind statistical hypothesis testing, consider the following example:

A case of suspected cheating on an exam is brought in front of the disciplinary committee at a certain university.

There are two opposing claims in this case:

• The student’s claim: I did not cheat on the exam.
• The instructor’s claim: The student did cheat on the exam.

Adhering to the principle “innocent until proven guilty,” the committee asks the instructor for evidence to support his claim. The instructor explains that the exam had two versions, and shows the committee members that on three separate exam questions, the student used in his solution numbers that were given in the other version of the exam.

The committee members all agree that it would be extremely unlikely to get evidence like that if the student’s claim of not cheating had been true. In other words, the committee members all agree that the instructor brought forward strong enough evidence to reject the student’s claim, and conclude that the student did cheat on the exam.

What does this example have to do with statistics?

While it is true that this story seems unrelated to statistics, it captures all the elements of hypothesis testing and the logic behind it. Before you read on to understand why, it would be useful to read the example again. Please do so now.

Statistical hypothesis testing is defined as:

• Assessing evidence provided by the data against the null claim (the claim which is to be assumed true unless enough evidence exists to reject it).

Here is how the process of statistical hypothesis testing works:

• We have two claims about what is going on in the population. Let’s call them claim 1 (this will be the null claim or hypothesis) and claim 2 (this will be the alternative) . Much like the story above, where the student’s claim is challenged by the instructor’s claim, the null claim 1 is challenged by the alternative claim 2. (For us, these claims are usually about the value of population parameter(s) or about the existence or nonexistence of a relationship between two variables in the population).
• We choose a sample, collect relevant data and summarize them (this is similar to the instructor collecting evidence from the student’s exam). For statistical tests, this step will also involve checking any conditions or assumptions.
• We figure out how likely it is to observe data like the data we obtained, if claim 1 is true. (Note that the wording “how likely …” implies that this step requires some kind of probability calculation). In the story, the committee members assessed how likely it is to observe evidence such as the instructor provided, had the student’s claim of not cheating been true.
• If, after assuming claim 1 is true, we find that it would be extremely unlikely to observe data as strong as ours or stronger in favor of claim 2, then we have strong evidence against claim 1, and we reject it in favor of claim 2. Later we will see this corresponds to a small p-value.
• If, after assuming claim 1 is true, we find that observing data as strong as ours or stronger in favor of claim 2 is NOT VERY UNLIKELY , then we do not have enough evidence against claim 1, and therefore we cannot reject it in favor of claim 2. Later we will see this corresponds to a p-value which is not small.

In our story, the committee decided that it would be extremely unlikely to find the evidence that the instructor provided had the student’s claim of not cheating been true. In other words, the members felt that it is extremely unlikely that it is just a coincidence (random chance) that the student used the numbers from the other version of the exam on three separate problems. The committee members therefore decided to reject the student’s claim and concluded that the student had, indeed, cheated on the exam. (Wouldn’t you conclude the same?)

Hopefully this example helped you understand the logic behind hypothesis testing.

Interactive Applet: Reasoning of a Statistical Test

To strengthen your understanding of the process of hypothesis testing and the logic behind it, let’s look at three statistical examples.

A recent study estimated that 20% of all college students in the United States smoke. The head of Health Services at Goodheart University (GU) suspects that the proportion of smokers may be lower at GU. In hopes of confirming her claim, the head of Health Services chooses a random sample of 400 Goodheart students, and finds that 70 of them are smokers.

Let’s analyze this example using the 4 steps outlined above:

• claim 1: The proportion of smokers at Goodheart is 0.20.
• claim 2: The proportion of smokers at Goodheart is less than 0.20.

Claim 1 basically says “nothing special goes on at Goodheart University; the proportion of smokers there is no different from the proportion in the entire country.” This claim is challenged by the head of Health Services, who suspects that the proportion of smokers at Goodheart is lower.

• Choosing a sample and collecting data: A sample of n = 400 was chosen, and summarizing the data revealed that the sample proportion of smokers is p -hat = 70/400 = 0.175.While it is true that 0.175 is less than 0.20, it is not clear whether this is strong enough evidence against claim 1. We must account for sampling variation.
• Assessment of evidence: In order to assess whether the data provide strong enough evidence against claim 1, we need to ask ourselves: How surprising is it to get a sample proportion as low as p -hat = 0.175 (or lower), assuming claim 1 is true? In other words, we need to find how likely it is that in a random sample of size n = 400 taken from a population where the proportion of smokers is p = 0.20 we’ll get a sample proportion as low as p -hat = 0.175 (or lower).It turns out that the probability that we’ll get a sample proportion as low as p -hat = 0.175 (or lower) in such a sample is roughly 0.106 (do not worry about how this was calculated at this point – however, if you think about it hopefully you can see that the key is the sampling distribution of p -hat).
• Conclusion: Well, we found that if claim 1 were true there is a probability of 0.106 of observing data like that observed or more extreme. Now you have to decide …Do you think that a probability of 0.106 makes our data rare enough (surprising enough) under claim 1 so that the fact that we did observe it is enough evidence to reject claim 1? Or do you feel that a probability of 0.106 means that data like we observed are not very likely when claim 1 is true, but they are not unlikely enough to conclude that getting such data is sufficient evidence to reject claim 1. Basically, this is your decision. However, it would be nice to have some kind of guideline about what is generally considered surprising enough.

A certain prescription allergy medicine is supposed to contain an average of 245 parts per million (ppm) of a certain chemical. If the concentration is higher than 245 ppm, the drug will likely cause unpleasant side effects, and if the concentration is below 245 ppm, the drug may be ineffective. The manufacturer wants to check whether the mean concentration in a large shipment is the required 245 ppm or not. To this end, a random sample of 64 portions from the large shipment is tested, and it is found that the sample mean concentration is 250 ppm with a sample standard deviation of 12 ppm.

• Claim 1: The mean concentration in the shipment is the required 245 ppm.
• Claim 2: The mean concentration in the shipment is not the required 245 ppm.

Note that again, claim 1 basically says: “There is nothing unusual about this shipment, the mean concentration is the required 245 ppm.” This claim is challenged by the manufacturer, who wants to check whether that is, indeed, the case or not.

• Choosing a sample and collecting data: A sample of n = 64 portions is chosen and after summarizing the data it is found that the sample mean concentration is x-bar = 250 and the sample standard deviation is s = 12.Is the fact that x-bar = 250 is different from 245 strong enough evidence to reject claim 1 and conclude that the mean concentration in the whole shipment is not the required 245? In other words, do the data provide strong enough evidence to reject claim 1?
• Assessing the evidence: In order to assess whether the data provide strong enough evidence against claim 1, we need to ask ourselves the following question: If the mean concentration in the whole shipment were really the required 245 ppm (i.e., if claim 1 were true), how surprising would it be to observe a sample of 64 portions where the sample mean concentration is off by 5 ppm or more (as we did)? It turns out that it would be extremely unlikely to get such a result if the mean concentration were really the required 245. There is only a probability of 0.0007 (i.e., 7 in 10,000) of that happening. (Do not worry about how this was calculated at this point, but again, the key will be the sampling distribution.)
• Making conclusions: Here, it is pretty clear that a sample like the one we observed or more extreme is VERY rare (or extremely unlikely) if the mean concentration in the shipment were really the required 245 ppm. The fact that we did observe such a sample therefore provides strong evidence against claim 1, so we reject it and conclude with very little doubt that the mean concentration in the shipment is not the required 245 ppm.

Do you think that you’re getting it? Let’s make sure, and look at another example.

Is there a relationship between gender and combined scores (Math + Verbal) on the SAT exam?

Following a report on the College Board website, which showed that in 2003, males scored generally higher than females on the SAT exam, an educational researcher wanted to check whether this was also the case in her school district. The researcher chose random samples of 150 males and 150 females from her school district, collected data on their SAT performance and found the following:

Again, let’s see how the process of hypothesis testing works for this example:

• Claim 1: Performance on the SAT is not related to gender (males and females score the same).
• Claim 2: Performance on the SAT is related to gender – males score higher.

Note that again, claim 1 basically says: “There is nothing going on between the variables SAT and gender.” Claim 2 represents what the researcher wants to check, or suspects might actually be the case.

• Choosing a sample and collecting data: Data were collected and summarized as given above. Is the fact that the sample mean score of males (1,025) is higher than the sample mean score of females (1,010) by 15 points strong enough information to reject claim 1 and conclude that in this researcher’s school district, males score higher on the SAT than females?
• Assessment of evidence: In order to assess whether the data provide strong enough evidence against claim 1, we need to ask ourselves: If SAT scores are in fact not related to gender (claim 1 is true), how likely is it to get data like the data we observed, in which the difference between the males’ average and females’ average score is as high as 15 points or higher? It turns out that the probability of observing such a sample result if SAT score is not related to gender is approximately 0.29 (Again, do not worry about how this was calculated at this point).
• Conclusion: Here, we have an example where observing a sample like the one we observed or more extreme is definitely not surprising (roughly 30% chance) if claim 1 were true (i.e., if indeed there is no difference in SAT scores between males and females). We therefore conclude that our data does not provide enough evidence for rejecting claim 1.
• “The data provide enough evidence to reject claim 1 and accept claim 2”; or
• “The data do not provide enough evidence to reject claim 1.”

In particular, note that in the second type of conclusion we did not say: “ I accept claim 1 ,” but only “ I don’t have enough evidence to reject claim 1 .” We will come back to this issue later, but this is a good place to make you aware of this subtle difference.

Hopefully by now, you understand the logic behind the statistical hypothesis testing process. Here is a summary:

Learn by Doing: Logic of Hypothesis Testing

Did I Get This?: Logic of Hypothesis Testing

Steps in Hypothesis Testing

Video: Steps in Hypothesis Testing (16:02)

Now that we understand the general idea of how statistical hypothesis testing works, let’s go back to each of the steps and delve slightly deeper, getting more details and learning some terminology.

Hypothesis Testing Step 1: State the Hypotheses

In all three examples, our aim is to decide between two opposing points of view, Claim 1 and Claim 2. In hypothesis testing, Claim 1 is called the null hypothesis (denoted “ Ho “), and Claim 2 plays the role of the alternative hypothesis (denoted “ Ha “). As we saw in the three examples, the null hypothesis suggests nothing special is going on; in other words, there is no change from the status quo, no difference from the traditional state of affairs, no relationship. In contrast, the alternative hypothesis disagrees with this, stating that something is going on, or there is a change from the status quo, or there is a difference from the traditional state of affairs. The alternative hypothesis, Ha, usually represents what we want to check or what we suspect is really going on.

Let’s go back to our three examples and apply the new notation:

In example 1:

• Ho: The proportion of smokers at GU is 0.20.
• Ha: The proportion of smokers at GU is less than 0.20.

In example 2:

• Ho: The mean concentration in the shipment is the required 245 ppm.
• Ha: The mean concentration in the shipment is not the required 245 ppm.

In example 3:

• Ho: Performance on the SAT is not related to gender (males and females score the same).
• Ha: Performance on the SAT is related to gender – males score higher.

Learn by Doing: State the Hypotheses

Did I Get This?: State the Hypotheses

Hypothesis Testing Step 2: Collect Data, Check Conditions and Summarize Data

This step is pretty obvious. This is what inference is all about. You look at sampled data in order to draw conclusions about the entire population. In the case of hypothesis testing, based on the data, you draw conclusions about whether or not there is enough evidence to reject Ho.

There is, however, one detail that we would like to add here. In this step we collect data and summarize it. Go back and look at the second step in our three examples. Note that in order to summarize the data we used simple sample statistics such as the sample proportion ( p -hat), sample mean (x-bar) and the sample standard deviation (s).

In practice, you go a step further and use these sample statistics to summarize the data with what’s called a test statistic . We are not going to go into any details right now, but we will discuss test statistics when we go through the specific tests.

This step will also involve checking any conditions or assumptions required to use the test.

Hypothesis Testing Step 3: Assess the Evidence

As we saw, this is the step where we calculate how likely is it to get data like that observed (or more extreme) when Ho is true. In a sense, this is the heart of the process, since we draw our conclusions based on this probability.

• If this probability is very small (see example 2), then that means that it would be very surprising to get data like that observed (or more extreme) if Ho were true. The fact that we did observe such data is therefore evidence against Ho, and we should reject it.
• On the other hand, if this probability is not very small (see example 3) this means that observing data like that observed (or more extreme) is not very surprising if Ho were true. The fact that we observed such data does not provide evidence against Ho. This crucial probability, therefore, has a special name. It is called the p-value of the test.

In our three examples, the p-values were given to you (and you were reassured that you didn’t need to worry about how these were derived yet):

• Example 1: p-value = 0.106
• Example 2: p-value = 0.0007
• Example 3: p-value = 0.29

Obviously, the smaller the p-value, the more surprising it is to get data like ours (or more extreme) when Ho is true, and therefore, the stronger the evidence the data provide against Ho.

Looking at the three p-values of our three examples, we see that the data that we observed in example 2 provide the strongest evidence against the null hypothesis, followed by example 1, while the data in example 3 provides the least evidence against Ho.

• Right now we will not go into specific details about p-value calculations, but just mention that since the p-value is the probability of getting data like those observed (or more extreme) when Ho is true, it would make sense that the calculation of the p-value will be based on the data summary, which, as we mentioned, is the test statistic. Indeed, this is the case. In practice, we will mostly use software to provide the p-value for us.

Hypothesis Testing Step 4: Making Conclusions

Since our statistical conclusion is based on how small the p-value is, or in other words, how surprising our data are when Ho is true, it would be nice to have some kind of guideline or cutoff that will help determine how small the p-value must be, or how “rare” (unlikely) our data must be when Ho is true, for us to conclude that we have enough evidence to reject Ho.

This cutoff exists, and because it is so important, it has a special name. It is called the significance level of the test and is usually denoted by the Greek letter α (alpha). The most commonly used significance level is α (alpha) = 0.05 (or 5%). This means that:

• if the p-value < α (alpha) (usually 0.05), then the data we obtained is considered to be “rare (or surprising) enough” under the assumption that Ho is true, and we say that the data provide statistically significant evidence against Ho, so we reject Ho and thus accept Ha.
• if the p-value > α (alpha)(usually 0.05), then our data are not considered to be “surprising enough” under the assumption that Ho is true, and we say that our data do not provide enough evidence to reject Ho (or, equivalently, that the data do not provide enough evidence to accept Ha).

Now that we have a cutoff to use, here are the appropriate conclusions for each of our examples based upon the p-values we were given.

In Example 1:

• Using our cutoff of 0.05, we fail to reject Ho.
• Conclusion : There IS NOT enough evidence that the proportion of smokers at GU is less than 0.20
• Still we should consider: Does the evidence seen in the data provide any practical evidence towards our alternative hypothesis?

In Example 2:

• Using our cutoff of 0.05, we reject Ho.
• Conclusion : There IS enough evidence that the mean concentration in the shipment is not the required 245 ppm.

In Example 3:

• Conclusion : There IS NOT enough evidence that males score higher on average than females on the SAT.

Notice that all of the above conclusions are written in terms of the alternative hypothesis and are given in the context of the situation. In no situation have we claimed the null hypothesis is true. Be very careful of this and other issues discussed in the following comments.

• Although the significance level provides a good guideline for drawing our conclusions, it should not be treated as an incontrovertible truth. There is a lot of room for personal interpretation. What if your p-value is 0.052? You might want to stick to the rules and say “0.052 > 0.05 and therefore I don’t have enough evidence to reject Ho”, but you might decide that 0.052 is small enough for you to believe that Ho should be rejected. It should be noted that scientific journals do consider 0.05 to be the cutoff point for which any p-value below the cutoff indicates enough evidence against Ho, and any p-value above it, or even equal to it , indicates there is not enough evidence against Ho. Although a p-value between 0.05 and 0.10 is often reported as marginally statistically significant.
• It is important to draw your conclusions in context . It is never enough to say: “p-value = …, and therefore I have enough evidence to reject Ho at the 0.05 significance level.” You should always word your conclusion in terms of the data. Although we will use the terminology of “rejecting Ho” or “failing to reject Ho” – this is mostly due to the fact that we are instructing you in these concepts. In practice, this language is rarely used. We also suggest writing your conclusion in terms of the alternative hypothesis.Is there or is there not enough evidence that the alternative hypothesis is true?
• Let’s go back to the issue of the nature of the two types of conclusions that I can make.
• Either I reject Ho (when the p-value is smaller than the significance level)
• or I cannot reject Ho (when the p-value is larger than the significance level).

As we mentioned earlier, note that the second conclusion does not imply that I accept Ho, but just that I don’t have enough evidence to reject it. Saying (by mistake) “I don’t have enough evidence to reject Ho so I accept it” indicates that the data provide evidence that Ho is true, which is not necessarily the case . Consider the following slightly artificial yet effective example:

An employer claims to subscribe to an “equal opportunity” policy, not hiring men any more often than women for managerial positions. Is this credible? You’re not sure, so you want to test the following two hypotheses:

• Ho: The proportion of male managers hired is 0.5
• Ha: The proportion of male managers hired is more than 0.5

Data: You choose at random three of the new managers who were hired in the last 5 years and find that all 3 are men.

Assessing Evidence: If the proportion of male managers hired is really 0.5 (Ho is true), then the probability that the random selection of three managers will yield three males is therefore 0.5 * 0.5 * 0.5 = 0.125. This is the p-value (using the multiplication rule for independent events).

Conclusion: Using 0.05 as the significance level, you conclude that since the p-value = 0.125 > 0.05, the fact that the three randomly selected managers were all males is not enough evidence to reject the employer’s claim of subscribing to an equal opportunity policy (Ho).

However, the data (all three selected are males) definitely does NOT provide evidence to accept the employer’s claim (Ho).

Learn By Doing: Using p-values

Did I Get This?: Using p-values

Comment about wording: Another common wording in scientific journals is:

• “The results are statistically significant” – when the p-value < α (alpha).
• “The results are not statistically significant” – when the p-value > α (alpha).

Often you will see significance levels reported with additional description to indicate the degree of statistical significance. A general guideline (although not required in our course) is:

• If 0.01 ≤ p-value < 0.05, then the results are (statistically) significant .
• If 0.001 ≤ p-value < 0.01, then the results are highly statistically significant .
• If p-value < 0.001, then the results are very highly statistically significant .
• If p-value > 0.05, then the results are not statistically significant (NS).
• If 0.05 ≤ p-value < 0.10, then the results are marginally statistically significant .

Let’s summarize

We learned quite a lot about hypothesis testing. We learned the logic behind it, what the key elements are, and what types of conclusions we can and cannot draw in hypothesis testing. Here is a quick recap:

Video: Hypothesis Testing Overview (2:20)

Here are a few more activities if you need some additional practice.

Did I Get This?: Hypothesis Testing Overview

• Notice that the p-value is an example of a conditional probability . We calculate the probability of obtaining results like those of our data (or more extreme) GIVEN the null hypothesis is true. We could write P(Obtaining results like ours or more extreme | Ho is True).
• We could write P(Obtaining a test statistic as or more extreme than ours | Ho is True).
• In this case we are asking “Assuming the null hypothesis is true, how rare is it to observe something as or more extreme than what I have found in my data?”
• If after assuming the null hypothesis is true, what we have found in our data is extremely rare (small p-value), this provides evidence to reject our assumption that Ho is true in favor of Ha.
• The p-value can also be thought of as the probability, assuming the null hypothesis is true, that the result we have seen is solely due to random error (or random chance). We have already seen that statistics from samples collected from a population vary. There is random error or random chance involved when we sample from populations.

In this setting, if the p-value is very small, this implies, assuming the null hypothesis is true, that it is extremely unlikely that the results we have obtained would have happened due to random error alone, and thus our assumption (Ho) is rejected in favor of the alternative hypothesis (Ha).

• It is EXTREMELY important that you find a definition of the p-value which makes sense to you. New students often need to contemplate this idea repeatedly through a variety of examples and explanations before becoming comfortable with this idea. It is one of the two most important concepts in statistics (the other being confidence intervals).
• We infer that the alternative hypothesis is true ONLY by rejecting the null hypothesis.
• A statistically significant result is one that has a very low probability of occurring if the null hypothesis is true.
• Results which are statistically significant may or may not have practical significance and vice versa.

Error and Power

LO 6.28: Define a Type I and Type II error in general and in the context of specific scenarios.

LO 6.29: Explain the concept of the power of a statistical test including the relationship between power, sample size, and effect size.

Video: Errors and Power (12:03)

Type I and Type II Errors in Hypothesis Tests

We have not yet discussed the fact that we are not guaranteed to make the correct decision by this process of hypothesis testing. Maybe you are beginning to see that there is always some level of uncertainty in statistics.

Let’s think about what we know already and define the possible errors we can make in hypothesis testing. When we conduct a hypothesis test, we choose one of two possible conclusions based upon our data.

If the p-value is smaller than your pre-specified significance level (α, alpha), you reject the null hypothesis and either

• You have made the correct decision since the null hypothesis is false
• You have made an error ( Type I ) and rejected Ho when in fact Ho is true (your data happened to be a RARE EVENT under Ho)

If the p-value is greater than (or equal to) your chosen significance level (α, alpha), you fail to reject the null hypothesis and either

• You have made the correct decision since the null hypothesis is true
• You have made an error ( Type II ) and failed to reject Ho when in fact Ho is false (the alternative hypothesis, Ha, is true)

The following summarizes the four possible results which can be obtained from a hypothesis test. Notice the rows represent the decision made in the hypothesis test and the columns represent the (usually unknown) truth in reality.

Although the truth is unknown in practice – or we would not be conducting the test – we know it must be the case that either the null hypothesis is true or the null hypothesis is false. It is also the case that either decision we make in a hypothesis test can result in an incorrect conclusion!

A TYPE I Error occurs when we Reject Ho when, in fact, Ho is True. In this case, we mistakenly reject a true null hypothesis.

• P(TYPE I Error) = P(Reject Ho | Ho is True) = α = alpha = Significance Level

A TYPE II Error occurs when we fail to Reject Ho when, in fact, Ho is False. In this case we fail to reject a false null hypothesis.

P(TYPE II Error) = P(Fail to Reject Ho | Ho is False) = β = beta

When our significance level is 5%, we are saying that we will allow ourselves to make a Type I error less than 5% of the time. In the long run, if we repeat the process, 5% of the time we will find a p-value < 0.05 when in fact the null hypothesis was true.

In this case, our data represent a rare occurrence which is unlikely to happen but is still possible. For example, suppose we toss a coin 10 times and obtain 10 heads, this is unlikely for a fair coin but not impossible. We might conclude the coin is unfair when in fact we simply saw a very rare event for this fair coin.

Our testing procedure CONTROLS for the Type I error when we set a pre-determined value for the significance level.

Notice that these probabilities are conditional probabilities. This is one more reason why conditional probability is an important concept in statistics.

Unfortunately, calculating the probability of a Type II error requires us to know the truth about the population. In practice we can only calculate this probability using a series of “what if” calculations which depend upon the type of problem.

Comment: As you initially read through the examples below, focus on the broad concepts instead of the small details. It is not important to understand how to calculate these values yourself at this point.

• Try to understand the pictures we present. Which pictures represent an assumed null hypothesis and which represent an alternative?
• It may be useful to come back to this page (and the activities here) after you have reviewed the rest of the section on hypothesis testing and have worked a few problems yourself.

Interactive Applet: Statistical Significance

Here are two examples of using an older version of this applet. It looks slightly different but the same settings and options are available in the version above.

In both cases we will consider IQ scores.

Our null hypothesis is that the true mean is 100. Assume the standard deviation is 16 and we will specify a significance level of 5%.

In this example we will specify that the true mean is indeed 100 so that the null hypothesis is true. Most of the time (95%), when we generate a sample, we should fail to reject the null hypothesis since the null hypothesis is indeed true.

Here is one sample that results in a correct decision:

In the sample above, we obtain an x-bar of 105, which is drawn on the distribution which assumes μ (mu) = 100 (the null hypothesis is true). Notice the sample is shown as blue dots along the x-axis and the shaded region shows for which values of x-bar we would reject the null hypothesis. In other words, we would reject Ho whenever the x-bar falls in the shaded region.

Enter the same values and generate samples until you obtain a Type I error (you falsely reject the null hypothesis). You should see something like this:

If you were to generate 100 samples, you should have around 5% where you rejected Ho. These would be samples which would result in a Type I error.

The previous example illustrates a correct decision and a Type I error when the null hypothesis is true. The next example illustrates a correct decision and Type II error when the null hypothesis is false. In this case, we must specify the true population mean.

Let’s suppose we are sampling from an honors program and that the true mean IQ for this population is 110. We do not know the probability of a Type II error without more detailed calculations.

Let’s start with a sample which results in a correct decision.

In the sample above, we obtain an x-bar of 111, which is drawn on the distribution which assumes μ (mu) = 100 (the null hypothesis is true).

Enter the same values and generate samples until you obtain a Type II error (you fail to reject the null hypothesis). You should see something like this:

You should notice that in this case (when Ho is false), it is easier to obtain an incorrect decision (a Type II error) than it was in the case where Ho is true. If you generate 100 samples, you can approximate the probability of a Type II error.

We can find the probability of a Type II error by visualizing both the assumed distribution and the true distribution together. The image below is adapted from an applet we will use when we discuss the power of a statistical test.

There is a 37.4% chance that, in the long run, we will make a Type II error and fail to reject the null hypothesis when in fact the true mean IQ is 110 in the population from which we sample our 10 individuals.

Can you visualize what will happen if the true population mean is really 115 or 108? When will the Type II error increase? When will it decrease? We will look at this idea again when we discuss the concept of power in hypothesis tests.

• It is important to note that there is a trade-off between the probability of a Type I and a Type II error. If we decrease the probability of one of these errors, the probability of the other will increase! The practical result of this is that if we require stronger evidence to reject the null hypothesis (smaller significance level = probability of a Type I error), we will increase the chance that we will be unable to reject the null hypothesis when in fact Ho is false (increases the probability of a Type II error).
• When α (alpha) = 0.05 we obtained a Type II error probability of 0.374 = β = beta

• When α (alpha) = 0.01 (smaller than before) we obtain a Type II error probability of 0.644 = β = beta (larger than before)

• As the blue line in the picture moves farther right, the significance level (α, alpha) is decreasing and the Type II error probability is increasing.
• As the blue line in the picture moves farther left, the significance level (α, alpha) is increasing and the Type II error probability is decreasing

Let’s return to our very first example and define these two errors in context.

• Ho = The student’s claim: I did not cheat on the exam.
• Ha = The instructor’s claim: The student did cheat on the exam.

Adhering to the principle “innocent until proven guilty,” the committee asks the instructor for evidence to support his claim.

There are four possible outcomes of this process. There are two possible correct decisions:

• The student did cheat on the exam and the instructor brings enough evidence to reject Ho and conclude the student did cheat on the exam. This is a CORRECT decision!
• The student did not cheat on the exam and the instructor fails to provide enough evidence that the student did cheat on the exam. This is a CORRECT decision!

Both the correct decisions and the possible errors are fairly easy to understand but with the errors, you must be careful to identify and define the two types correctly.

TYPE I Error: Reject Ho when Ho is True

• The student did not cheat on the exam but the instructor brings enough evidence to reject Ho and conclude the student cheated on the exam. This is a Type I Error.

TYPE II Error: Fail to Reject Ho when Ho is False

• The student did cheat on the exam but the instructor fails to provide enough evidence that the student cheated on the exam. This is a Type II Error.

In most situations, including this one, it is more “acceptable” to have a Type II error than a Type I error. Although allowing a student who cheats to go unpunished might be considered a very bad problem, punishing a student for something he or she did not do is usually considered to be a more severe error. This is one reason we control for our Type I error in the process of hypothesis testing.

Did I Get This?: Type I and Type II Errors (in context)

• The probabilities of Type I and Type II errors are closely related to the concepts of sensitivity and specificity that we discussed previously. Consider the following hypotheses:

Ho: The individual does not have diabetes (status quo, nothing special happening)

Ha: The individual does have diabetes (something is going on here)

In this setting:

When someone tests positive for diabetes we would reject the null hypothesis and conclude the person has diabetes (we may or may not be correct!).

When someone tests negative for diabetes we would fail to reject the null hypothesis so that we fail to conclude the person has diabetes (we may or may not be correct!)

Let’s take it one step further:

Sensitivity = P(Test + | Have Disease) which in this setting equals P(Reject Ho | Ho is False) = 1 – P(Fail to Reject Ho | Ho is False) = 1 – β = 1 – beta

Specificity = P(Test – | No Disease) which in this setting equals P(Fail to Reject Ho | Ho is True) = 1 – P(Reject Ho | Ho is True) = 1 – α = 1 – alpha

Notice that sensitivity and specificity relate to the probability of making a correct decision whereas α (alpha) and β (beta) relate to the probability of making an incorrect decision.

Usually α (alpha) = 0.05 so that the specificity listed above is 0.95 or 95%.

Next, we will see that the sensitivity listed above is the power of the hypothesis test!

Reasons for a Type I Error in Practice

Assuming that you have obtained a quality sample:

• The reason for a Type I error is random chance.
• When a Type I error occurs, our observed data represented a rare event which indicated evidence in favor of the alternative hypothesis even though the null hypothesis was actually true.

Reasons for a Type II Error in Practice

Again, assuming that you have obtained a quality sample, now we have a few possibilities depending upon the true difference that exists.

• The sample size is too small to detect an important difference. This is the worst case, you should have obtained a larger sample. In this situation, you may notice that the effect seen in the sample seems PRACTICALLY significant and yet the p-value is not small enough to reject the null hypothesis.
• The sample size is reasonable for the important difference but the true difference (which might be somewhat meaningful or interesting) is smaller than your test was capable of detecting. This is tolerable as you were not interested in being able to detect this difference when you began your study. In this situation, you may notice that the effect seen in the sample seems to have some potential for practical significance.
• The sample size is more than adequate, the difference that was not detected is meaningless in practice. This is not a problem at all and is in effect a “correct decision” since the difference you did not detect would have no practical meaning.
• Note: We will discuss the idea of practical significance later in more detail.

Power of a Hypothesis Test

It is often the case that we truly wish to prove the alternative hypothesis. It is reasonable that we would be interested in the probability of correctly rejecting the null hypothesis. In other words, the probability of rejecting the null hypothesis, when in fact the null hypothesis is false. This can also be thought of as the probability of being able to detect a (pre-specified) difference of interest to the researcher.

Let’s begin with a realistic example of how power can be described in a study.

In a clinical trial to study two medications for weight loss, we have an 80% chance to detect a difference in the weight loss between the two medications of 10 pounds. In other words, the power of the hypothesis test we will conduct is 80%.

In other words, if one medication comes from a population with an average weight loss of 25 pounds and the other comes from a population with an average weight loss of 15 pounds, we will have an 80% chance to detect that difference using the sample we have in our trial.

If we were to repeat this trial many times, 80% of the time we will be able to reject the null hypothesis (that there is no difference between the medications) and 20% of the time we will fail to reject the null hypothesis (and make a Type II error!).

The difference of 10 pounds in the previous example, is often called the effect size . The measure of the effect differs depending on the particular test you are conducting but is always some measure related to the true effect in the population. In this example, it is the difference between two population means.

Recall the definition of a Type II error:

Notice that P(Reject Ho | Ho is False) = 1 – P(Fail to Reject Ho | Ho is False) = 1 – β = 1- beta.

The POWER of a hypothesis test is the probability of rejecting the null hypothesis when the null hypothesis is false . This can also be stated as the probability of correctly rejecting the null hypothesis .

POWER = P(Reject Ho | Ho is False) = 1 – β = 1 – beta

Power is the test’s ability to correctly reject the null hypothesis. A test with high power has a good chance of being able to detect the difference of interest to us, if it exists .

As we mentioned on the bottom of the previous page, this can be thought of as the sensitivity of the hypothesis test if you imagine Ho = No disease and Ha = Disease.

Factors Affecting the Power of a Hypothesis Test

The power of a hypothesis test is affected by numerous quantities (similar to the margin of error in a confidence interval).

Assume that the null hypothesis is false for a given hypothesis test. All else being equal, we have the following:

• Larger samples result in a greater chance to reject the null hypothesis which means an increase in the power of the hypothesis test.
• If the effect size is larger, it will become easier for us to detect. This results in a greater chance to reject the null hypothesis which means an increase in the power of the hypothesis test. The effect size varies for each test and is usually closely related to the difference between the hypothesized value and the true value of the parameter under study.
• From the relationship between the probability of a Type I and a Type II error (as α (alpha) decreases, β (beta) increases), we can see that as α (alpha) decreases, Power = 1 – β = 1 – beta also decreases.
• There are other mathematical ways to change the power of a hypothesis test, such as changing the population standard deviation; however, these are not quantities that we can usually control so we will not discuss them here.

In practice, we specify a significance level and a desired power to detect a difference which will have practical meaning to us and this determines the sample size required for the experiment or study.

For most grants involving statistical analysis, power calculations must be completed to illustrate that the study will have a reasonable chance to detect an important effect. Otherwise, the money spent on the study could be wasted. The goal is usually to have a power close to 80%.

For example, if there is only a 5% chance to detect an important difference between two treatments in a clinical trial, this would result in a waste of time, effort, and money on the study since, when the alternative hypothesis is true, the chance a treatment effect can be found is very small.

• In order to calculate the power of a hypothesis test, we must specify the “truth.” As we mentioned previously when discussing Type II errors, in practice we can only calculate this probability using a series of “what if” calculations which depend upon the type of problem.

The following activity involves working with an interactive applet to study power more carefully.

Learn by Doing: Power of Hypothesis Tests

The following reading is an excellent discussion about Type I and Type II errors.

(Optional) Outside Reading: A Good Discussion of Power (≈ 2500 words)

We will not be asking you to perform power calculations manually. You may be asked to use online calculators and applets. Most statistical software packages offer some ability to complete power calculations. There are also many online calculators for power and sample size on the internet, for example, Russ Lenth’s power and sample-size page .

Proportions (Introduction & Step 1)

CO-4: Distinguish among different measurement scales, choose the appropriate descriptive and inferential statistical methods based on these distinctions, and interpret the results.

LO 4.33: In a given context, distinguish between situations involving a population proportion and a population mean and specify the correct null and alternative hypothesis for the scenario.

LO 4.34: Carry out a complete hypothesis test for a population proportion by hand.

Video: Proportions (Introduction & Step 1) (7:18)

Now that we understand the process of hypothesis testing and the logic behind it, we are ready to start learning about specific statistical tests (also known as significance tests).

The first test we are going to learn is the test about the population proportion (p).

This test is widely known as the “z-test for the population proportion (p).”

We will understand later where the “z-test” part is coming from.

This will be the only type of problem you will complete entirely “by-hand” in this course. Our goal is to use this example to give you the tools you need to understand how this process works. After working a few problems, you should review the earlier material again. You will likely need to review the terminology and concepts a few times before you fully understand the process.

In reality, you will often be conducting more complex statistical tests and allowing software to provide the p-value. In these settings it will be important to know what test to apply for a given situation and to be able to explain the results in context.

Review: Types of Variables

When we conduct a test about a population proportion, we are working with a categorical variable. Later in the course, after we have learned a variety of hypothesis tests, we will need to be able to identify which test is appropriate for which situation. Identifying the variable as categorical or quantitative is an important component of choosing an appropriate hypothesis test.

Learn by Doing: Review Types of Variables

One Sample Z-Test for a Population Proportion

In this part of our discussion on hypothesis testing, we will go into details that we did not go into before. More specifically, we will use this test to introduce the idea of a test statistic , and details about how p-values are calculated .

Let’s start by introducing the three examples, which will be the leading examples in our discussion. Each example is followed by a figure illustrating the information provided, as well as the question of interest.

A machine is known to produce 20% defective products, and is therefore sent for repair. After the machine is repaired, 400 products produced by the machine are chosen at random and 64 of them are found to be defective. Do the data provide enough evidence that the proportion of defective products produced by the machine (p) has been reduced as a result of the repair?

The following figure displays the information, as well as the question of interest:

The question of interest helps us formulate the null and alternative hypotheses in terms of p, the proportion of defective products produced by the machine following the repair:

• Ho: p = 0.20 (No change; the repair did not help).
• Ha: p < 0.20 (The repair was effective at reducing the proportion of defective parts).

There are rumors that students at a certain liberal arts college are more inclined to use drugs than U.S. college students in general. Suppose that in a simple random sample of 100 students from the college, 19 admitted to marijuana use. Do the data provide enough evidence to conclude that the proportion of marijuana users among the students in the college (p) is higher than the national proportion, which is 0.157? (This number is reported by the Harvard School of Public Health.)

Again, the following figure displays the information as well as the question of interest:

As before, we can formulate the null and alternative hypotheses in terms of p, the proportion of students in the college who use marijuana:

• Ho: p = 0.157 (same as among all college students in the country).
• Ha: p > 0.157 (higher than the national figure).

Polls on certain topics are conducted routinely in order to monitor changes in the public’s opinions over time. One such topic is the death penalty. In 2003 a poll estimated that 64% of U.S. adults support the death penalty for a person convicted of murder. In a more recent poll, 675 out of 1,000 U.S. adults chosen at random were in favor of the death penalty for convicted murderers. Do the results of this poll provide evidence that the proportion of U.S. adults who support the death penalty for convicted murderers (p) changed between 2003 and the later poll?

Here is a figure that displays the information, as well as the question of interest:

Again, we can formulate the null and alternative hypotheses in term of p, the proportion of U.S. adults who support the death penalty for convicted murderers.

• Ho: p = 0.64 (No change from 2003).
• Ha: p ≠ 0.64 (Some change since 2003).

Learn by Doing: Proportions (Overview)

Did I Get This?: Proportions ( Overview )

Recall that there are basically 4 steps in the process of hypothesis testing:

• STEP 1: State the appropriate null and alternative hypotheses, Ho and Ha.
• STEP 2: Obtain a random sample, collect relevant data, and check whether the data meet the conditions under which the test can be used . If the conditions are met, summarize the data using a test statistic.
• STEP 3: Find the p-value of the test.
• STEP 4: Based on the p-value, decide whether or not the results are statistically significant and draw your conclusions in context.
• Note: In practice, we should always consider the practical significance of the results as well as the statistical significance.

We are now going to go through these steps as they apply to the hypothesis testing for the population proportion p. It should be noted that even though the details will be specific to this particular test, some of the ideas that we will add apply to hypothesis testing in general.

Step 1. Stating the Hypotheses

Here again are the three set of hypotheses that are being tested in each of our three examples:

Has the proportion of defective products been reduced as a result of the repair?

Is the proportion of marijuana users in the college higher than the national figure?

Did the proportion of U.S. adults who support the death penalty change between 2003 and a later poll?

The null hypothesis always takes the form:

• Ho: p = some value

and the alternative hypothesis takes one of the following three forms:

• Ha: p < that value (like in example 1) or
• Ha: p > that value (like in example 2) or
• Ha: p ≠ that value (like in example 3).

Note that it was quite clear from the context which form of the alternative hypothesis would be appropriate. The value that is specified in the null hypothesis is called the null value , and is generally denoted by p 0 . We can say, therefore, that in general the null hypothesis about the population proportion (p) would take the form:

• Ho: p = p 0

We write Ho: p = p 0 to say that we are making the hypothesis that the population proportion has the value of p 0 . In other words, p is the unknown population proportion and p 0 is the number we think p might be for the given situation.

The alternative hypothesis takes one of the following three forms (depending on the context):

Ha: p < p 0 (one-sided)

Ha: p > p 0 (one-sided)

Ha: p ≠ p 0 (two-sided)

The first two possible forms of the alternatives (where the = sign in Ho is challenged by < or >) are called one-sided alternatives , and the third form of alternative (where the = sign in Ho is challenged by ≠) is called a two-sided alternative. To understand the intuition behind these names let’s go back to our examples.

Example 3 (death penalty) is a case where we have a two-sided alternative:

In this case, in order to reject Ho and accept Ha we will need to get a sample proportion of death penalty supporters which is very different from 0.64 in either direction, either much larger or much smaller than 0.64.

In example 2 (marijuana use) we have a one-sided alternative:

Here, in order to reject Ho and accept Ha we will need to get a sample proportion of marijuana users which is much higher than 0.157.

Similarly, in example 1 (defective products), where we are testing:

in order to reject Ho and accept Ha, we will need to get a sample proportion of defective products which is much smaller than 0.20.

Learn by Doing: State Hypotheses (Proportions)

Did I Get This?: State Hypotheses (Proportions)

Proportions (Step 2)

Video: Proportions (Step 2) (12:38)

Step 2. Collect Data, Check Conditions, and Summarize Data

After the hypotheses have been stated, the next step is to obtain a sample (on which the inference will be based), collect relevant data , and summarize them.

It is extremely important that our sample is representative of the population about which we want to draw conclusions. This is ensured when the sample is chosen at random. Beyond the practical issue of ensuring representativeness, choosing a random sample has theoretical importance that we will mention later.

In the case of hypothesis testing for the population proportion (p), we will collect data on the relevant categorical variable from the individuals in the sample and start by calculating the sample proportion p-hat (the natural quantity to calculate when the parameter of interest is p).

Let’s go back to our three examples and add this step to our figures.

As we mentioned earlier without going into details, when we summarize the data in hypothesis testing, we go a step beyond calculating the sample statistic and summarize the data with a test statistic . Every test has a test statistic, which to some degree captures the essence of the test. In fact, the p-value, which so far we have looked upon as “the king” (in the sense that everything is determined by it), is actually determined by (or derived from) the test statistic. We will now introduce the test statistic.

The test statistic is a measure of how far the sample proportion p-hat is from the null value p 0 , the value that the null hypothesis claims is the value of p. In other words, since p-hat is what the data estimates p to be, the test statistic can be viewed as a measure of the “distance” between what the data tells us about p and what the null hypothesis claims p to be.

Let’s use our examples to understand this:

The parameter of interest is p, the proportion of defective products following the repair.

The data estimate p to be p-hat = 0.16

The null hypothesis claims that p = 0.20

The data are therefore 0.04 (or 4 percentage points) below the null hypothesis value.

It is hard to evaluate whether this difference of 4% in defective products is enough evidence to say that the repair was effective at reducing the proportion of defective products, but clearly, the larger the difference, the more evidence it is against the null hypothesis. So if, for example, our sample proportion of defective products had been, say, 0.10 instead of 0.16, then I think you would all agree that cutting the proportion of defective products in half (from 20% to 10%) would be extremely strong evidence that the repair was effective at reducing the proportion of defective products.

The parameter of interest is p, the proportion of students in a college who use marijuana.

The data estimate p to be p-hat = 0.19

The null hypothesis claims that p = 0.157

The data are therefore 0.033 (or 3.3. percentage points) above the null hypothesis value.

The parameter of interest is p, the proportion of U.S. adults who support the death penalty for convicted murderers.

The data estimate p to be p-hat = 0.675

The null hypothesis claims that p = 0.64

There is a difference of 0.035 (or 3.5. percentage points) between the data and the null hypothesis value.

The problem with looking only at the difference between the sample proportion, p-hat, and the null value, p 0 is that we have not taken into account the variability of our estimator p-hat which, as we know from our study of sampling distributions, depends on the sample size.

For this reason, the test statistic cannot simply be the difference between p-hat and p 0 , but must be some form of that formula that accounts for the sample size. In other words, we need to somehow standardize the difference so that comparison between different situations will be possible. We are very close to revealing the test statistic, but before we construct it, let’s be reminded of the following two facts from probability:

Fact 1: When we take a random sample of size n from a population with population proportion p, then

Fact 2: The z-score of any normal value (a value that comes from a normal distribution) is calculated by finding the difference between the value and the mean and then dividing that difference by the standard deviation (of the normal distribution associated with the value). The z-score represents how many standard deviations below or above the mean the value is.

Thus, our test statistic should be a measure of how far the sample proportion p-hat is from the null value p 0 relative to the variation of p-hat (as measured by the standard error of p-hat).

Recall that the standard error is the standard deviation of the sampling distribution for a given statistic. For p-hat, we know the following:

To find the p-value, we will need to determine how surprising our value is assuming the null hypothesis is true. We already have the tools needed for this process from our study of sampling distributions as represented in the table above.

If we assume the null hypothesis is true, we can specify that the center of the distribution of all possible values of p-hat from samples of size 400 would be 0.20 (our null value).

We can calculate the standard error, assuming p = 0.20 as

\(\sqrt{\dfrac{p_{0}\left(1-p_{0}\right)}{n}}=\sqrt{\dfrac{0.2(1-0.2)}{400}}=0.02\)

The following picture represents the sampling distribution of all possible values of p-hat of samples of size 400, assuming the true proportion p is 0.20 and our other requirements for the sampling distribution to be normal are met (we will review these during the next step).

In order to calculate probabilities for the picture above, we would need to find the z-score associated with our result.

This z-score is the test statistic ! In this example, the numerator of our z-score is the difference between p-hat (0.16) and null value (0.20) which we found earlier to be -0.04. The denominator of our z-score is the standard error calculated above (0.02) and thus quickly we find the z-score, our test statistic, to be -2.

The sample proportion based upon this data is 2 standard errors below the null value.

Hopefully you now understand more about the reasons we need probability in statistics!!

Now we will formalize the definition and look at our remaining examples before moving on to the next step, which will be to determine if a normal distribution applies and calculate the p-value.

Test Statistic for Hypothesis Tests for One Proportion is:

\(z=\dfrac{\hat{p}-p_{0}}{\sqrt{\dfrac{p_{0}\left(1-p_{0}\right)}{n}}}\)

It represents the difference between the sample proportion and the null value, measured in standard deviations (standard error of p-hat).

The picture above is a representation of the sampling distribution of p-hat assuming p = p 0 . In other words, this is a model of how p-hat behaves if we are drawing random samples from a population for which Ho is true.

Notice the center of the sampling distribution is at p 0 , which is the hypothesized proportion given in the null hypothesis (Ho: p = p 0 .) We could also mark the axis in standard error units,

\(\sqrt{\dfrac{p_{0}\left(1-p_{0}\right)}{n}}\)

For example, if our null hypothesis claims that the proportion of U.S. adults supporting the death penalty is 0.64, then the sampling distribution is drawn as if the null is true. We draw a normal distribution centered at 0.64 (p 0 ) with a standard error dependent on sample size,

\(\sqrt{\dfrac{0.64(1-0.64)}{n}}\).

Important Comment:

• Note that under the assumption that Ho is true (and if the conditions for the sampling distribution to be normal are satisfied) the test statistic follows a N(0,1) (standard normal) distribution. Another way to say the same thing which is quite common is: “The null distribution of the test statistic is N(0,1).”

By “null distribution,” we mean the distribution under the assumption that Ho is true. As we’ll see and stress again later, the null distribution of the test statistic is what the calculation of the p-value is based on.

Let’s go back to our remaining two examples and find the test statistic in each case:

Since the null hypothesis is Ho: p = 0.157, the standardized (z) score of p-hat = 0.19 is

\(z=\dfrac{0.19-0.157}{\sqrt{\dfrac{0.157(1-0.157)}{100}}} \approx 0.91\)

This is the value of the test statistic for this example.

We interpret this to mean that, assuming that Ho is true, the sample proportion p-hat = 0.19 is 0.91 standard errors above the null value (0.157).

Since the null hypothesis is Ho: p = 0.64, the standardized (z) score of p-hat = 0.675 is

\(z=\dfrac{0.675-0.64}{\sqrt{\dfrac{0.64(1-0.64)}{1000}}} \approx 2.31\)

We interpret this to mean that, assuming that Ho is true, the sample proportion p-hat = 0.675 is 2.31 standard errors above the null value (0.64).

Learn by Doing: Proportions (Step 2)

Comments about the Test Statistic:

• We mentioned earlier that to some degree, the test statistic captures the essence of the test. In this case, the test statistic measures the difference between p-hat and p 0 in standard errors. This is exactly what this test is about. Get data, and look at the discrepancy between what the data estimates p to be (represented by p-hat) and what Ho claims about p (represented by p 0 ).
• You can think about this test statistic as a measure of evidence in the data against Ho. The larger the test statistic, the “further the data are from Ho” and therefore the more evidence the data provide against Ho.

Learn by Doing: Proportions (Step 2) Understanding the Test Statistic

Did I Get This?: Proportions (Step 2)

• It should now be clear why this test is commonly known as the z-test for the population proportion . The name comes from the fact that it is based on a test statistic that is a z-score.
• Recall fact 1 that we used for constructing the z-test statistic. Here is part of it again:

When we take a random sample of size n from a population with population proportion p 0 , the possible values of the sample proportion p-hat ( when certain conditions are met ) have approximately a normal distribution with a mean of p 0 … and a standard deviation of

This result provides the theoretical justification for constructing the test statistic the way we did, and therefore the assumptions under which this result holds (in bold, above) are the conditions that our data need to satisfy so that we can use this test. These two conditions are:

i. The sample has to be random.

ii. The conditions under which the sampling distribution of p-hat is normal are met. In other words:

• Here we will pause to say more about condition (i.) above, the need for a random sample. In the Probability Unit we discussed sampling plans based on probability (such as a simple random sample, cluster, or stratified sampling) that produce a non-biased sample, which can be safely used in order to make inferences about a population. We noted in the Probability Unit that, in practice, other (non-random) sampling techniques are sometimes used when random sampling is not feasible. It is important though, when these techniques are used, to be aware of the type of bias that they introduce, and thus the limitations of the conclusions that can be drawn from them. For our purpose here, we will focus on one such practice, the situation in which a sample is not really chosen randomly, but in the context of the categorical variable that is being studied, the sample is regarded as random. For example, say that you are interested in the proportion of students at a certain college who suffer from seasonal allergies. For that purpose, the students in a large engineering class could be considered as a random sample, since there is nothing about being in an engineering class that makes you more or less likely to suffer from seasonal allergies. Technically, the engineering class is a convenience sample, but it is treated as a random sample in the context of this categorical variable. On the other hand, if you are interested in the proportion of students in the college who have math anxiety, then the class of engineering students clearly could not possibly be viewed as a random sample, since engineering students probably have a much lower incidence of math anxiety than the college population overall.

Learn by Doing: Proportions (Step 2) Valid or Invalid Sampling?

Let’s check the conditions in our three examples.

i. The 400 products were chosen at random.

ii. n = 400, p 0 = 0.2 and therefore:

\(n p_{0}=400(0.2)=80 \geq 10\)

\(n\left(1-p_{0}\right)=400(1-0.2)=320 \geq 10\)

i. The 100 students were chosen at random.

ii. n = 100, p 0 = 0.157 and therefore:

\begin{gathered} n p_{0}=100(0.157)=15.7 \geq 10 \\ n\left(1-p_{0}\right)=100(1-0.157)=84.3 \geq 10 \end{gathered}

i. The 1000 adults were chosen at random.

ii. n = 1000, p 0 = 0.64 and therefore:

\begin{gathered} n p_{0}=1000(0.64)=640 \geq 10 \\ n\left(1-p_{0}\right)=1000(1-0.64)=360 \geq 10 \end{gathered}

Learn by Doing: Proportions (Step 2) Verify Conditions

Checking that our data satisfy the conditions under which the test can be reliably used is a very important part of the hypothesis testing process. Be sure to consider this for every hypothesis test you conduct in this course and certainly in practice.

The Four Steps in Hypothesis Testing

With respect to the z-test, the population proportion that we are currently discussing we have:

Step 1: Completed

Step 2: Completed

Step 3: This is what we will work on next.

Proportions (Step 3)

Video: Proportions (Step 3) (14:46)

Calculators and Tables

Step 3. Finding the P-value of the Test

So far we’ve talked about the p-value at the intuitive level: understanding what it is (or what it measures) and how we use it to draw conclusions about the statistical significance of our results. We will now go more deeply into how the p-value is calculated.

It should be mentioned that eventually we will rely on technology to calculate the p-value for us (as well as the test statistic), but in order to make intelligent use of the output, it is important to first understand the details, and only then let the computer do the calculations for us. Again, our goal is to use this simple example to give you the tools you need to understand the process entirely. Let’s start.

Recall that so far we have said that the p-value is the probability of obtaining data like those observed assuming that Ho is true. Like the test statistic, the p-value is, therefore, a measure of the evidence against Ho. In the case of the test statistic, the larger it is in magnitude (positive or negative), the further p-hat is from p 0 , the more evidence we have against Ho. In the case of the p-value , it is the opposite; the smaller it is, the more unlikely it is to get data like those observed when Ho is true, the more evidence it is against Ho . One can actually draw conclusions in hypothesis testing just using the test statistic, and as we’ll see the p-value is, in a sense, just another way of looking at the test statistic. The reason that we actually take the extra step in this course and derive the p-value from the test statistic is that even though in this case (the test about the population proportion) and some other tests, the value of the test statistic has a very clear and intuitive interpretation, there are some tests where its value is not as easy to interpret. On the other hand, the p-value keeps its intuitive appeal across all statistical tests.

How is the p-value calculated?

Intuitively, the p-value is the probability of observing data like those observed assuming that Ho is true. Let’s be a bit more formal:

• Since this is a probability question about the data , it makes sense that the calculation will involve the data summary, the test statistic.
• What do we mean by “like” those observed? By “like” we mean “as extreme or even more extreme.”

Putting it all together, we get that in general:

The p-value is the probability of observing a test statistic as extreme as that observed (or even more extreme) assuming that the null hypothesis is true.

By “extreme” we mean extreme in the direction(s) of the alternative hypothesis.

Specifically , for the z-test for the population proportion:

• If the alternative hypothesis is Ha: p < p 0 (less than) , then “extreme” means small or less than , and the p-value is: The probability of observing a test statistic as small as that observed or smaller if the null hypothesis is true.
• If the alternative hypothesis is Ha: p > p 0 (greater than) , then “extreme” means large or greater than , and the p-value is: The probability of observing a test statistic as large as that observed or larger if the null hypothesis is true.
• If the alternative is Ha: p ≠ p 0 (different from) , then “extreme” means extreme in either direction either small or large (i.e., large in magnitude) or just different from , and the p-value therefore is: The probability of observing a test statistic as large in magnitude as that observed or larger if the null hypothesis is true.(Examples: If z = -2.5: p-value = probability of observing a test statistic as small as -2.5 or smaller or as large as 2.5 or larger. If z = 1.5: p-value = probability of observing a test statistic as large as 1.5 or larger, or as small as -1.5 or smaller.)

OK, hopefully that makes (some) sense. But how do we actually calculate it?

Recall the important comment from our discussion about our test statistic,

which said that when the null hypothesis is true (i.e., when p = p 0 ), the possible values of our test statistic follow a standard normal (N(0,1), denoted by Z) distribution. Therefore, the p-value calculations (which assume that Ho is true) are simply standard normal distribution calculations for the 3 possible alternative hypotheses.

Alternative Hypothesis is “Less Than”

The probability of observing a test statistic as small as that observed or smaller , assuming that the values of the test statistic follow a standard normal distribution. We will now represent this probability in symbols and also using the normal distribution.

Looking at the shaded region, you can see why this is often referred to as a left-tailed test. We shaded to the left of the test statistic, since less than is to the left.

Alternative Hypothesis is “Greater Than”

The probability of observing a test statistic as large as that observed or larger , assuming that the values of the test statistic follow a standard normal distribution. Again, we will represent this probability in symbols and using the normal distribution

Looking at the shaded region, you can see why this is often referred to as a right-tailed test. We shaded to the right of the test statistic, since greater than is to the right.

Alternative Hypothesis is “Not Equal To”

The probability of observing a test statistic which is as large in magnitude as that observed or larger, assuming that the values of the test statistic follow a standard normal distribution.

This is often referred to as a two-tailed test, since we shaded in both directions.

Next, we will apply this to our three examples. But first, work through the following activities, which should help your understanding.

Learn by Doing: Proportions (Step 3)

Did I Get This?: Proportions (Step 3)

The p-value in this case is:

• The probability of observing a test statistic as small as -2 or smaller, assuming that Ho is true.

OR (recalling what the test statistic actually means in this case),

• The probability of observing a sample proportion that is 2 standard deviations or more below the null value (p 0 = 0.20), assuming that p 0 is the true population proportion.

OR, more specifically,

• The probability of observing a sample proportion of 0.16 or lower in a random sample of size 400, when the true population proportion is p 0 =0.20

In either case, the p-value is found as shown in the following figure:

To find P(Z ≤ -2) we can either use the calculator or table we learned to use in the probability unit for normal random variables. Eventually, after we understand the details, we will use software to run the test for us and the output will give us all the information we need. The p-value that the statistical software provides for this specific example is 0.023. The p-value tells us that it is pretty unlikely (probability of 0.023) to get data like those observed (test statistic of -2 or less) assuming that Ho is true.

• The probability of observing a test statistic as large as 0.91 or larger, assuming that Ho is true.
• The probability of observing a sample proportion that is 0.91 standard deviations or more above the null value (p 0 = 0.157), assuming that p 0 is the true population proportion.
• The probability of observing a sample proportion of 0.19 or higher in a random sample of size 100, when the true population proportion is p 0 =0.157

Again, at this point we can either use the calculator or table to find that the p-value is 0.182, this is P(Z ≥ 0.91).

The p-value tells us that it is not very surprising (probability of 0.182) to get data like those observed (which yield a test statistic of 0.91 or higher) assuming that the null hypothesis is true.

• The probability of observing a test statistic as large as 2.31 (or larger) or as small as -2.31 (or smaller), assuming that Ho is true.
• The probability of observing a sample proportion that is 2.31 standard deviations or more away from the null value (p 0 = 0.64), assuming that p 0 is the true population proportion.
• The probability of observing a sample proportion as different as 0.675 is from 0.64, or even more different (i.e. as high as 0.675 or higher or as low as 0.605 or lower) in a random sample of size 1,000, when the true population proportion is p 0 = 0.64

Again, at this point we can either use the calculator or table to find that the p-value is 0.021, this is P(Z ≤ -2.31) + P(Z ≥ 2.31) = 2*P(Z ≥ |2.31|)

The p-value tells us that it is pretty unlikely (probability of 0.021) to get data like those observed (test statistic as high as 2.31 or higher or as low as -2.31 or lower) assuming that Ho is true.

• We’ve just seen that finding p-values involves probability calculations about the value of the test statistic assuming that Ho is true. In this case, when Ho is true, the values of the test statistic follow a standard normal distribution (i.e., the sampling distribution of the test statistic when the null hypothesis is true is N(0,1)). Therefore, p-values correspond to areas (probabilities) under the standard normal curve.

Similarly, in any test , p-values are found using the sampling distribution of the test statistic when the null hypothesis is true (also known as the “null distribution” of the test statistic). In this case, it was relatively easy to argue that the null distribution of our test statistic is N(0,1). As we’ll see, in other tests, other distributions come up (like the t-distribution and the F-distribution), which we will just mention briefly, and rely heavily on the output of our statistical package for obtaining the p-values.

We’ve just completed our discussion about the p-value, and how it is calculated both in general and more specifically for the z-test for the population proportion. Let’s go back to the four-step process of hypothesis testing and see what we’ve covered and what still needs to be discussed.

With respect to the z-test the population proportion:

Step 3: Completed

Step 4. This is what we will work on next.

Learn by Doing: Proportions (Step 3) Understanding P-values

Proportions (Step 4 & Summary)

Video: Proportions (Step 4 & Summary) (4:30)

Step 4. Drawing Conclusions Based on the P-Value

This last part of the four-step process of hypothesis testing is the same across all statistical tests, and actually, we’ve already said basically everything there is to say about it, but it can’t hurt to say it again.

The p-value is a measure of how much evidence the data present against Ho. The smaller the p-value, the more evidence the data present against Ho.

We already mentioned that what determines what constitutes enough evidence against Ho is the significance level (α, alpha), a cutoff point below which the p-value is considered small enough to reject Ho in favor of Ha. The most commonly used significance level is 0.05.

• Conclusion: There IS enough evidence that Ha is True
• Conclusion: There IS NOT enough evidence that Ha is True

Where instead of Ha is True , we write what this means in the words of the problem, in other words, in the context of the current scenario.

It is important to mention again that this step has essentially two sub-steps:

(i) Based on the p-value, determine whether or not the results are statistically significant (i.e., the data present enough evidence to reject Ho).

(ii) State your conclusions in the context of the problem.

Note: We always still must consider whether the results have any practical significance, particularly if they are statistically significant as a statistically significant result which has not practical use is essentially meaningless!

Let’s go back to our three examples and draw conclusions.

We found that the p-value for this test was 0.023.

Since 0.023 is small (in particular, 0.023 < 0.05), the data provide enough evidence to reject Ho.

Conclusion:

• There IS enough evidence that the proportion of defective products is less than 20% after the repair .

The following figure is the complete story of this example, and includes all the steps we went through, starting from stating the hypotheses and ending with our conclusions:

We found that the p-value for this test was 0.182.

Since .182 is not small (in particular, 0.182 > 0.05), the data do not provide enough evidence to reject Ho.

• There IS NOT enough evidence that the proportion of students at the college who use marijuana is higher than the national figure.

Here is the complete story of this example:

Learn by Doing: Learn by Doing – Proportions (Step 4)

We found that the p-value for this test was 0.021.

Since 0.021 is small (in particular, 0.021 < 0.05), the data provide enough evidence to reject Ho

• There IS enough evidence that the proportion of adults who support the death penalty for convicted murderers has changed since 2003.

Did I Get This?: Proportions (Step 4)

Many Students Wonder: Hypothesis Testing for the Population Proportion

Many students wonder why 5% is often selected as the significance level in hypothesis testing, and why 1% is the next most typical level. This is largely due to just convenience and tradition.

When Ronald Fisher (one of the founders of modern statistics) published one of his tables, he used a mathematically convenient scale that included 5% and 1%. Later, these same 5% and 1% levels were used by other people, in part just because Fisher was so highly esteemed. But mostly these are arbitrary levels.

The idea of selecting some sort of relatively small cutoff was historically important in the development of statistics; but it’s important to remember that there is really a continuous range of increasing confidence towards the alternative hypothesis, not a single all-or-nothing value. There isn’t much meaningful difference, for instance, between a p-value of .049 or .051, and it would be foolish to declare one case definitely a “real” effect and to declare the other case definitely a “random” effect. In either case, the study results were roughly 5% likely by chance if there’s no actual effect.

Whether such a p-value is sufficient for us to reject a particular null hypothesis ultimately depends on the risk of making the wrong decision, and the extent to which the hypothesized effect might contradict our prior experience or previous studies.

Let’s Summarize!!

We have now completed going through the four steps of hypothesis testing, and in particular we learned how they are applied to the z-test for the population proportion. Here is a brief summary:

Step 1: State the hypotheses

State the null hypothesis:

State the alternative hypothesis:

where the choice of the appropriate alternative (out of the three) is usually quite clear from the context of the problem. If you feel it is not clear, it is most likely a two-sided problem. Students are usually good at recognizing the “more than” and “less than” terminology but differences can sometimes be more difficult to spot, sometimes this is because you have preconceived ideas of how you think it should be! Use only the information given in the problem.

Step 2: Obtain data, check conditions, and summarize data

Obtain data from a sample and:

(i) Check whether the data satisfy the conditions which allow you to use this test.

random sample (or at least a sample that can be considered random in context)

the conditions under which the sampling distribution of p-hat is normal are met

(ii) Calculate the sample proportion p-hat, and summarize the data using the test statistic:

( Recall: This standardized test statistic represents how many standard deviations above or below p 0 our sample proportion p-hat is.)

Step 3: Find the p-value of the test by using the test statistic as follows

IMPORTANT FACT: In all future tests, we will rely on software to obtain the p-value.

When the alternative hypothesis is “less than” the probability of observing a test statistic as small as that observed or smaller , assuming that the values of the test statistic follow a standard normal distribution. We will now represent this probability in symbols and also using the normal distribution.

When the alternative hypothesis is “greater than” the probability of observing a test statistic as large as that observed or larger , assuming that the values of the test statistic follow a standard normal distribution. Again, we will represent this probability in symbols and using the normal distribution

When the alternative hypothesis is “not equal to” the probability of observing a test statistic which is as large in magnitude as that observed or larger, assuming that the values of the test statistic follow a standard normal distribution.

Step 4: Conclusion

Reach a conclusion first regarding the statistical significance of the results, and then determine what it means in the context of the problem.

If p-value ≤ 0.05 then WE REJECT Ho Conclusion: There IS enough evidence that Ha is True

If p-value > 0.05 then WE FAIL TO REJECT Ho Conclusion: There IS NOT enough evidence that Ha is True

Recall that: If the p-value is small (in particular, smaller than the significance level, which is usually 0.05), the results are statistically significant (in the sense that there is a statistically significant difference between what was observed in the sample and what was claimed in Ho), and so we reject Ho.

If the p-value is not small, we do not have enough statistical evidence to reject Ho, and so we continue to believe that Ho may be true. ( Remember: In hypothesis testing we never “accept” Ho ).

Finally, in practice, we should always consider the practical significance of the results as well as the statistical significance.

Learn by Doing: Z-Test for a Population Proportion

What’s next?

Before we move on to the next test, we are going to use the z-test for proportions to bring up and illustrate a few more very important issues regarding hypothesis testing. This might also be a good time to review the concepts of Type I error, Type II error, and Power before continuing on.

More about Hypothesis Testing

CO-1: Describe the roles biostatistics serves in the discipline of public health.

LO 1.11: Recognize the distinction between statistical significance and practical significance.

LO 6.30: Use a confidence interval to determine the correct conclusion to the associated two-sided hypothesis test.

Video: More about Hypothesis Testing (18:25)

The issues regarding hypothesis testing that we will discuss are:

• The effect of sample size on hypothesis testing.
• Statistical significance vs. practical importance.
• Hypothesis testing and confidence intervals—how are they related?

Let’s begin.

1. The Effect of Sample Size on Hypothesis Testing

We have already seen the effect that the sample size has on inference, when we discussed point and interval estimation for the population mean (μ, mu) and population proportion (p). Intuitively …

Larger sample sizes give us more information to pin down the true nature of the population. We can therefore expect the sample mean and sample proportion obtained from a larger sample to be closer to the population mean and proportion, respectively. As a result, for the same level of confidence, we can report a smaller margin of error, and get a narrower confidence interval. What we’ve seen, then, is that larger sample size gives a boost to how much we trust our sample results.

In hypothesis testing, larger sample sizes have a similar effect. We have also discussed that the power of our test increases when the sample size increases, all else remaining the same. This means, we have a better chance to detect the difference between the true value and the null value for larger samples.

The following two examples will illustrate that a larger sample size provides more convincing evidence (the test has greater power), and how the evidence manifests itself in hypothesis testing. Let’s go back to our example 2 (marijuana use at a certain liberal arts college).

We do not have enough evidence to conclude that the proportion of students at the college who use marijuana is higher than the national figure.

Now, let’s increase the sample size.

There are rumors that students in a certain liberal arts college are more inclined to use drugs than U.S. college students in general. Suppose that in a simple random sample of 400 students from the college, 76 admitted to marijuana use . Do the data provide enough evidence to conclude that the proportion of marijuana users among the students in the college (p) is higher than the national proportion, which is 0.157? (Reported by the Harvard School of Public Health).

Our results here are statistically significant . In other words, in example 2* the data provide enough evidence to reject Ho.

• Conclusion: There is enough evidence that the proportion of marijuana users at the college is higher than among all U.S. students.

What do we learn from this?

We see that sample results that are based on a larger sample carry more weight (have greater power).

In example 2, we saw that a sample proportion of 0.19 based on a sample of size of 100 was not enough evidence that the proportion of marijuana users in the college is higher than 0.157. Recall, from our general overview of hypothesis testing, that this conclusion (not having enough evidence to reject the null hypothesis) doesn’t mean the null hypothesis is necessarily true (so, we never “accept” the null); it only means that the particular study didn’t yield sufficient evidence to reject the null. It might be that the sample size was simply too small to detect a statistically significant difference.

However, in example 2*, we saw that when the sample proportion of 0.19 is obtained from a sample of size 400, it carries much more weight, and in particular, provides enough evidence that the proportion of marijuana users in the college is higher than 0.157 (the national figure). In this case, the sample size of 400 was large enough to detect a statistically significant difference.

The following activity will allow you to practice the ideas and terminology used in hypothesis testing when a result is not statistically significant.

Learn by Doing: Interpreting Non-significant Results

2. Statistical significance vs. practical importance.

Now, we will address the issue of statistical significance versus practical importance (which also involves issues of sample size).

The following activity will let you explore the effect of the sample size on the statistical significance of the results yourself, and more importantly will discuss issue 2: Statistical significance vs. practical importance.

Important Fact: In general, with a sufficiently large sample size you can make any result that has very little practical importance statistically significant! A large sample size alone does NOT make a “good” study!!

This suggests that when interpreting the results of a test, you should always think not only about the statistical significance of the results but also about their practical importance.

Learn by Doing: Statistical vs. Practical Significance

3. Hypothesis Testing and Confidence Intervals

The last topic we want to discuss is the relationship between hypothesis testing and confidence intervals. Even though the flavor of these two forms of inference is different (confidence intervals estimate a parameter, and hypothesis testing assesses the evidence in the data against one claim and in favor of another), there is a strong link between them.

We will explain this link (using the z-test and confidence interval for the population proportion), and then explain how confidence intervals can be used after a test has been carried out.

Recall that a confidence interval gives us a set of plausible values for the unknown population parameter. We may therefore examine a confidence interval to informally decide if a proposed value of population proportion seems plausible.

For example, if a 95% confidence interval for p, the proportion of all U.S. adults already familiar with Viagra in May 1998, was (0.61, 0.67), then it seems clear that we should be able to reject a claim that only 50% of all U.S. adults were familiar with the drug, since based on the confidence interval, 0.50 is not one of the plausible values for p.

In fact, the information provided by a confidence interval can be formally related to the information provided by a hypothesis test. ( Comment: The relationship is more straightforward for two-sided alternatives, and so we will not present results for the one-sided cases.)

Suppose we want to carry out the two-sided test:

• Ha: p ≠ p 0

using a significance level of 0.05.

An alternative way to perform this test is to find a 95% confidence interval for p and check:

• If p 0 falls outside the confidence interval, reject Ho.
• If p 0 falls inside the confidence interval, do not reject Ho.

In other words,

• If p 0 is not one of the plausible values for p, we reject Ho.
• If p 0 is a plausible value for p, we cannot reject Ho.

( Comment: Similarly, the results of a test using a significance level of 0.01 can be related to the 99% confidence interval.)

Let’s look at an example:

Recall example 3, where we wanted to know whether the proportion of U.S. adults who support the death penalty for convicted murderers has changed since 2003, when it was 0.64.

We are testing:

and as the figure reminds us, we took a sample of 1,000 U.S. adults, and the data told us that 675 supported the death penalty for convicted murderers (p-hat = 0.675).

A 95% confidence interval for p, the proportion of all U.S. adults who support the death penalty, is:

\(0.675 \pm 1.96 \sqrt{\dfrac{0.675(1-0.675)}{1000}} \approx 0.675 \pm 0.029=(0.646,0.704)\)

Since the 95% confidence interval for p does not include 0.64 as a plausible value for p, we can reject Ho and conclude (as we did before) that there is enough evidence that the proportion of U.S. adults who support the death penalty for convicted murderers has changed since 2003.

You and your roommate are arguing about whose turn it is to clean the apartment. Your roommate suggests that you settle this by tossing a coin and takes one out of a locked box he has on the shelf. Suspecting that the coin might not be fair, you decide to test it first. You toss the coin 80 times, thinking to yourself that if, indeed, the coin is fair, you should get around 40 heads. Instead you get 48 heads. You are puzzled. You are not sure whether getting 48 heads out of 80 is enough evidence to conclude that the coin is unbalanced, or whether this a result that could have happened just by chance when the coin is fair.

Statistics can help you answer this question.

Let p be the true proportion (probability) of heads. We want to test whether the coin is fair or not.

• Ho: p = 0.5 (the coin is fair).
• Ha: p ≠ 0.5 (the coin is not fair).

The data we have are that out of n = 80 tosses, we got 48 heads, or that the sample proportion of heads is p-hat = 48/80 = 0.6.

A 95% confidence interval for p, the true proportion of heads for this coin, is:

\(0.6 \pm 1.96 \sqrt{\dfrac{0.6(1-0.6)}{80}} \approx 0.6 \pm 0.11=(0.49,0.71)\)

Since in this case 0.5 is one of the plausible values for p, we cannot reject Ho. In other words, the data do not provide enough evidence to conclude that the coin is not fair.

The context of the last example is a good opportunity to bring up an important point that was discussed earlier.

Even though we use 0.05 as a cutoff to guide our decision about whether the results are statistically significant, we should not treat it as inviolable and we should always add our own judgment. Let’s look at the last example again.

It turns out that the p-value of this test is 0.0734. In other words, it is maybe not extremely unlikely, but it is quite unlikely (probability of 0.0734) that when you toss a fair coin 80 times you’ll get a sample proportion of heads of 48/80 = 0.6 (or even more extreme). It is true that using the 0.05 significance level (cutoff), 0.0734 is not considered small enough to conclude that the coin is not fair. However, if you really don’t want to clean the apartment, the p-value might be small enough for you to ask your roommate to use a different coin, or to provide one yourself!

Did I Get This?: Connection between Confidence Intervals and Hypothesis Tests

Did I Get This?: Hypothesis Tests for Proportions (Extra Practice)

Here is our final point on this subject:

When the data provide enough evidence to reject Ho, we can conclude (depending on the alternative hypothesis) that the population proportion is either less than, greater than, or not equal to the null value p 0 . However, we do not get a more informative statement about its actual value. It might be of interest, then, to follow the test with a 95% confidence interval that will give us more insight into the actual value of p.

In our example 3,

we concluded that the proportion of U.S. adults who support the death penalty for convicted murderers has changed since 2003, when it was 0.64. It is probably of interest not only to know that the proportion has changed, but also to estimate what it has changed to. We’ve calculated the 95% confidence interval for p on the previous page and found that it is (0.646, 0.704).

We can combine our conclusions from the test and the confidence interval and say:

Data provide evidence that the proportion of U.S. adults who support the death penalty for convicted murderers has changed since 2003, and we are 95% confident that it is now between 0.646 and 0.704. (i.e. between 64.6% and 70.4%).

Let’s look at our example 1 to see how a confidence interval following a test might be insightful in a different way.

Here is a summary of example 1:

We conclude that as a result of the repair, the proportion of defective products has been reduced to below 0.20 (which was the proportion prior to the repair). It is probably of great interest to the company not only to know that the proportion of defective has been reduced, but also estimate what it has been reduced to, to get a better sense of how effective the repair was. A 95% confidence interval for p in this case is:

\(0.16 \pm 1.96 \sqrt{\dfrac{0.16(1-0.16)}{400}} \approx 0.16 \pm 0.036=(0.124,0.196)\)

We can therefore say that the data provide evidence that the proportion of defective products has been reduced, and we are 95% confident that it has been reduced to somewhere between 12.4% and 19.6%. This is very useful information, since it tells us that even though the results were significant (i.e., the repair reduced the number of defective products), the repair might not have been effective enough, if it managed to reduce the number of defective products only to the range provided by the confidence interval. This, of course, ties back in to the idea of statistical significance vs. practical importance that we discussed earlier. Even though the results are statistically significant (Ho was rejected), practically speaking, the repair might still be considered ineffective.

Learn by Doing: Hypothesis Tests and Confidence Intervals

Even though this portion of the current section is about the z-test for population proportion, it is loaded with very important ideas that apply to hypothesis testing in general. We’ve already summarized the details that are specific to the z-test for proportions, so the purpose of this summary is to highlight the general ideas.

The process of hypothesis testing has four steps :

I. Stating the null and alternative hypotheses (Ho and Ha).

II. Obtaining a random sample (or at least one that can be considered random) and collecting data. Using the data:

Check that the conditions under which the test can be reliably used are met.

Summarize the data using a test statistic.

• The test statistic is a measure of the evidence in the data against Ho. The larger the test statistic is in magnitude, the more evidence the data present against Ho.

III. Finding the p-value of the test. The p-value is the probability of getting data like those observed (or even more extreme) assuming that the null hypothesis is true, and is calculated using the null distribution of the test statistic. The p-value is a measure of the evidence against Ho. The smaller the p-value, the more evidence the data present against Ho.

IV. Making conclusions.

Conclusions about the statistical significance of the results:

If the p-value is small, the data present enough evidence to reject Ho (and accept Ha).

If the p-value is not small, the data do not provide enough evidence to reject Ho.

To help guide our decision, we use the significance level as a cutoff for what is considered a small p-value. The significance cutoff is usually set at 0.05.

Conclusions should then be provided in the context of the problem.

Additional Important Ideas about Hypothesis Testing

• Results that are based on a larger sample carry more weight, and therefore as the sample size increases, results become more statistically significant.
• Even a very small and practically unimportant effect becomes statistically significant with a large enough sample size. The distinction between statistical significance and practical importance should therefore always be considered.
• Confidence intervals can be used in order to carry out two-sided tests (95% confidence for the 0.05 significance level). If the null value is not included in the confidence interval (i.e., is not one of the plausible values for the parameter), we have enough evidence to reject Ho. Otherwise, we cannot reject Ho.
• If the results are statistically significant, it might be of interest to follow up the tests with a confidence interval in order to get insight into the actual value of the parameter of interest.
• It is important to be aware that there are two types of errors in hypothesis testing ( Type I and Type II ) and that the power of a statistical test is an important measure of how likely we are to be able to detect a difference of interest to us in a particular problem.

Means (All Steps)

NOTE: Beginning on this page, the Learn By Doing and Did I Get This activities are presented as interactive PDF files. The interactivity may not work on mobile devices or with certain PDF viewers. Use an official ADOBE product such as ADOBE READER .

If you have any issues with the Learn By Doing or Did I Get This interactive PDF files, you can view all of the questions and answers presented on this page in this document:

• QUESTION/Answer (SPOILER ALERT!)

Tests About μ (mu) When σ (sigma) is Unknown – The t-test for a Population Mean

The t-distribution.

Video: Means (All Steps) (13:11)

So far we have talked about the logic behind hypothesis testing and then illustrated how this process proceeds in practice, using the z-test for the population proportion (p).

We are now moving on to discuss testing for the population mean (μ, mu), which is the parameter of interest when the variable of interest is quantitative.

A few comments about the structure of this section:

• The basic groundwork for carrying out hypothesis tests has already been laid in our general discussion and in our presentation of tests about proportions.

Therefore we can easily modify the four steps to carry out tests about means instead, without going into all of the details again.

We will use this approach for all future tests so be sure to go back to the discussion in general and for proportions to review the concepts in more detail.

• In our discussion about confidence intervals for the population mean, we made the distinction between whether the population standard deviation, σ (sigma) was known or if we needed to estimate this value using the sample standard deviation, s .

In this section, we will only discuss the second case as in most realistic settings we do not know the population standard deviation .

In this case we need to use the t- distribution instead of the standard normal distribution for the probability aspects of confidence intervals (choosing table values) and hypothesis tests (finding p-values).

• Although we will discuss some theoretical or conceptual details for some of the analyses we will learn, from this point on we will rely on software to conduct tests and calculate confidence intervals for us , while we focus on understanding which methods are used for which situations and what the results say in context.

If you are interested in more information about the z-test, where we assume the population standard deviation σ (sigma) is known, you can review the Carnegie Mellon Open Learning Statistics Course (you will need to click “ENTER COURSE”).

Like any other tests, the t- test for the population mean follows the four-step process:

• STEP 1: Stating the hypotheses H o and H a .
• STEP 2: Collecting relevant data, checking that the data satisfy the conditions which allow us to use this test, and summarizing the data using a test statistic.
• STEP 3: Finding the p-value of the test, the probability of obtaining data as extreme as those collected (or even more extreme, in the direction of the alternative hypothesis), assuming that the null hypothesis is true. In other words, how likely is it that the only reason for getting data like those observed is sampling variability (and not because H o is not true)?
• STEP 4: Drawing conclusions, assessing the statistical significance of the results based on the p-value, and stating our conclusions in context. (Do we or don’t we have evidence to reject H o and accept H a ?)
• Note: In practice, we should also always consider the practical significance of the results as well as the statistical significance.

We will now go through the four steps specifically for the t- test for the population mean and apply them to our two examples.

Only in a few cases is it reasonable to assume that the population standard deviation, σ (sigma), is known and so we will not cover hypothesis tests in this case. We discussed both cases for confidence intervals so that we could still calculate some confidence intervals by hand.

For this and all future tests we will rely on software to obtain our summary statistics, test statistics, and p-values for us.

The case where σ (sigma) is unknown is much more common in practice. What can we use to replace σ (sigma)? If you don’t know the population standard deviation, the best you can do is find the sample standard deviation, s, and use it instead of σ (sigma). (Note that this is exactly what we did when we discussed confidence intervals).

Is that it? Can we just use s instead of σ (sigma), and the rest is the same as the previous case? Unfortunately, it’s not that simple, but not very complicated either.

Here, when we use the sample standard deviation, s, as our estimate of σ (sigma) we can no longer use a normal distribution to find the cutoff for confidence intervals or the p-values for hypothesis tests.

Instead we must use the t- distribution (with n-1 degrees of freedom) to obtain the p-value for this test.

We discussed this issue for confidence intervals. We will talk more about the t- distribution after we discuss the details of this test for those who are interested in learning more.

It isn’t really necessary for us to understand this distribution but it is important that we use the correct distributions in practice via our software.

We will wait until UNIT 4B to look at how to accomplish this test in the software. For now focus on understanding the process and drawing the correct conclusions from the p-values given.

Now let’s go through the four steps in conducting the t- test for the population mean.

The null and alternative hypotheses for the t- test for the population mean (μ, mu) have exactly the same structure as the hypotheses for z-test for the population proportion (p):

The null hypothesis has the form:

• Ho: μ = μ 0 (mu = mu_zero)

(where μ 0 (mu_zero) is often called the null value)

• Ha: μ < μ 0 (mu < mu_zero) (one-sided)
• Ha: μ > μ 0 (mu > mu_zero) (one-sided)
• Ha: μ ≠ μ 0 (mu ≠ mu_zero) (two-sided)

where the choice of the appropriate alternative (out of the three) is usually quite clear from the context of the problem.

If you feel it is not clear, it is most likely a two-sided problem. Students are usually good at recognizing the “more than” and “less than” terminology but differences can sometimes be more difficult to spot, sometimes this is because you have preconceived ideas of how you think it should be! You also cannot use the information from the sample to help you determine the hypothesis. We would not know our data when we originally asked the question.

Now try it yourself. Here are a few exercises on stating the hypotheses for tests for a population mean.

Learn by Doing: State the Hypotheses for a test for a population mean

Here are a few more activities for practice.

Did I Get This?: State the Hypotheses for a test for a population mean

When setting up hypotheses, be sure to use only the information in the research question. We cannot use our sample data to help us set up our hypotheses.

For this test, it is still important to correctly choose the alternative hypothesis as “less than”, “greater than”, or “different” although generally in practice two-sample tests are used.

Obtain data from a sample:

• In this step we would obtain data from a sample. This is not something we do much of in courses but it is done very often in practice!

Check the conditions:

• Then we check the conditions under which this test (the t- test for one population mean) can be safely carried out – which are:
• The sample is random (or at least can be considered random in context).
• We are in one of the three situations marked with a green check mark in the following table (which ensure that x-bar is at least approximately normal and the test statistic using the sample standard deviation, s, is therefore a t- distribution with n-1 degrees of freedom – proving this is beyond the scope of this course):
• For large samples, we don’t need to check for normality in the population . We can rely on the sample size as the basis for the validity of using this test.
• For small samples , we need to have data from a normal population in order for the p-values and confidence intervals to be valid.

In practice, for small samples, it can be very difficult to determine if the population is normal. Here is a simulation to give you a better understanding of the difficulties.

Video: Simulations – Are Samples from a Normal Population? (4:58)

Now try it yourself with a few activities.

Learn by Doing: Checking Conditions for Hypothesis Testing for the Population Mean

• It is always a good idea to look at the data and get a sense of their pattern regardless of whether you actually need to do it in order to assess whether the conditions are met.
• This idea of looking at the data is relevant to all tests in general. In the next module—inference for relationships—conducting exploratory data analysis before inference will be an integral part of the process.

Here are a few more problems for extra practice.

Did I Get This?: Checking Conditions for Hypothesis Testing for the Population Mean

When setting up hypotheses, be sure to use only the information in the res

Calculate Test Statistic

Assuming that the conditions are met, we calculate the sample mean x-bar and the sample standard deviation, s (which estimates σ (sigma)), and summarize the data with a test statistic.

The test statistic for the t -test for the population mean is:

\(t=\dfrac{\bar{x} - \mu_0}{s/ \sqrt{n}}\)

Recall that such a standardized test statistic represents how many standard deviations above or below μ 0 (mu_zero) our sample mean x-bar is.

Therefore our test statistic is a measure of how different our data are from what is claimed in the null hypothesis. This is an idea that we mentioned in the previous test as well.

Again we will rely on the p-value to determine how unusual our data would be if the null hypothesis is true.

As we mentioned, the test statistic in the t -test for a population mean does not follow a standard normal distribution. Rather, it follows another bell-shaped distribution called the t- distribution.

We will present the details of this distribution at the end for those interested but for now we will work on the process of the test.

Here are a few important facts.

• In statistical language we say that the null distribution of our test statistic is the t- distribution with (n-1) degrees of freedom. In other words, when Ho is true (i.e., when μ = μ 0 (mu = mu_zero)), our test statistic has a t- distribution with (n-1) d.f., and this is the distribution under which we find p-values.
• For a large sample size (n), the null distribution of the test statistic is approximately Z, so whether we use t (n – 1) or Z to calculate the p-values does not make a big difference. However, software will use the t -distribution regardless of the sample size and so will we.

Although we will not calculate p-values by hand for this test, we can still easily calculate the test statistic.

Try it yourself:

Learn by Doing: Calculate the Test Statistic for a Test for a Population Mean

From this point in this course and certainly in practice we will allow the software to calculate our test statistics and we will use the p-values provided to draw our conclusions.

We will use software to obtain the p-value for this (and all future) tests but here are the images illustrating how the p-value is calculated in each of the three cases corresponding to the three choices for our alternative hypothesis.

Note that due to the symmetry of the t distribution, for a given value of the test statistic t, the p-value for the two-sided test is twice as large as the p-value of either of the one-sided tests. The same thing happens when p-values are calculated under the t distribution as when they are calculated under the Z distribution.

We will show some examples of p-values obtained from software in our examples. For now let’s continue our summary of the steps.

As usual, based on the p-value (and some significance level of choice) we assess the statistical significance of results, and draw our conclusions in context.

To review what we have said before:

If p-value ≤ 0.05 then WE REJECT Ho

If p-value > 0.05 then WE FAIL TO REJECT Ho

This step has essentially two sub-steps:

We are now ready to look at two examples.

A certain prescription medicine is supposed to contain an average of 250 parts per million (ppm) of a certain chemical. If the concentration is higher than this, the drug may cause harmful side effects; if it is lower, the drug may be ineffective.

The manufacturer runs a check to see if the mean concentration in a large shipment conforms to the target level of 250 ppm or not.

A simple random sample of 100 portions is tested, and the sample mean concentration is found to be 247 ppm with a sample standard deviation of 12 ppm.

Here is a figure that represents this example:

1. The hypotheses being tested are:

• Ha: μ ≠ μ 0 (mu ≠ mu_zero)
• Where μ = population mean part per million of the chemical in the entire shipment

2. The conditions that allow us to use the t-test are met since:

• The sample is random
• The sample size is large enough for the Central Limit Theorem to apply and ensure the normality of x-bar. We do not need normality of the population in order to be able to conduct this test for the population mean. We are in the 2 nd column in the table below.
• The test statistic is:

\(t=\dfrac{\bar{x}-\mu_{0}}{s / \sqrt{n}}=\dfrac{247-250}{12 / \sqrt{100}}=-2.5\)

• The data (represented by the sample mean) are 2.5 standard errors below the null value.

3. Finding the p-value.

• To find the p-value we use statistical software, and we calculate a p-value of 0.014.

4. Conclusions:

• The p-value is small (.014) indicating that at the 5% significance level, the results are significant.
• We reject the null hypothesis.
• There is enough evidence to conclude that the mean concentration in entire shipment is not the required 250 ppm.
• It is difficult to comment on the practical significance of this result without more understanding of the practical considerations of this problem.

Here is a summary:

• The 95% confidence interval for μ (mu) can be used here in the same way as for proportions to conduct the two-sided test (checking whether the null value falls inside or outside the confidence interval) or following a t- test where Ho was rejected to get insight into the value of μ (mu).
• We find the 95% confidence interval to be (244.619, 249.381) . Since 250 is not in the interval we know we would reject our null hypothesis that μ (mu) = 250. The confidence interval gives additional information. By accounting for estimation error, it estimates that the population mean is likely to be between 244.62 and 249.38. This is lower than the target concentration and that information might help determine the seriousness and appropriate course of action in this situation.

In most situations in practice we use TWO-SIDED HYPOTHESIS TESTS, followed by confidence intervals to gain more insight.

For completeness in covering one sample t-tests for a population mean, we still cover all three possible alternative hypotheses here HOWEVER, this will be the last test for which we will do so.

A research study measured the pulse rates of 57 college men and found a mean pulse rate of 70 beats per minute with a standard deviation of 9.85 beats per minute.

Researchers want to know if the mean pulse rate for all college men is different from the current standard of 72 beats per minute.

• The hypotheses being tested are:
• Ho: μ = 72
• Ha: μ ≠ 72
• Where μ = population mean heart rate among college men
• The conditions that allow us to use the t- test are met since:
• The sample is random.
• The sample size is large (n = 57) so we do not need normality of the population in order to be able to conduct this test for the population mean. We are in the 2 nd column in the table below.

\(t=\dfrac{\bar{x}-\mu}{s / \sqrt{n}}=\dfrac{70-72}{9.85 / \sqrt{57}}=-1.53\)

• The data (represented by the sample mean) are 1.53 estimated standard errors below the null value.
• Recall that in general the p-value is calculated under the null distribution of the test statistic, which, in the t- test case, is t (n-1). In our case, in which n = 57, the p-value is calculated under the t (56) distribution. Using statistical software, we find that the p-value is 0.132 .
• Here is how we calculated the p-value. http://homepage.stat.uiowa.edu/~mbognar/applets/t.html .

4. Making conclusions.

• The p-value (0.132) is not small, indicating that the results are not significant.
• We fail to reject the null hypothesis.
• There is not enough evidence to conclude that the mean pulse rate for all college men is different from the current standard of 72 beats per minute.
• The results from this sample do not appear to have any practical significance either with a mean pulse rate of 70, this is very similar to the hypothesized value, relative to the variation expected in pulse rates.

Now try a few yourself.

Learn by Doing: Hypothesis Testing for the Population Mean

From this point in this course and certainly in practice we will allow the software to calculate our test statistic and p-value and we will use the p-values provided to draw our conclusions.

That concludes our discussion of hypothesis tests in Unit 4A.

In the next unit we will continue to use both confidence intervals and hypothesis test to investigate the relationship between two variables in the cases we covered in Unit 1 on exploratory data analysis – we will look at Case CQ, Case CC, and Case QQ.

Before moving on, we will discuss the details about the t- distribution as a general object.

We have seen that variables can be visually modeled by many different sorts of shapes, and we call these shapes distributions. Several distributions arise so frequently that they have been given special names, and they have been studied mathematically.

So far in the course, the only one we’ve named, for continuous quantitative variables, is the normal distribution, but there are others. One of them is called the t- distribution.

The t- distribution is another bell-shaped (unimodal and symmetric) distribution, like the normal distribution; and the center of the t- distribution is standardized at zero, like the center of the standard normal distribution.

Like all distributions that are used as probability models, the normal and the t- distribution are both scaled, so the total area under each of them is 1.

So how is the t-distribution fundamentally different from the normal distribution?

• The spread .

The following picture illustrates the fundamental difference between the normal distribution and the t-distribution:

Here we have an image which illustrates the fundamental difference between the normal distribution and the t- distribution:

You can see in the picture that the t- distribution has slightly less area near the expected central value than the normal distribution does, and you can see that the t distribution has correspondingly more area in the “tails” than the normal distribution does. (It’s often said that the t- distribution has “fatter tails” or “heavier tails” than the normal distribution.)

This reflects the fact that the t- distribution has a larger spread than the normal distribution. The same total area of 1 is spread out over a slightly wider range on the t- distribution, making it a bit lower near the center compared to the normal distribution, and giving the t- distribution slightly more probability in the ‘tails’ compared to the normal distribution.

Therefore, the t- distribution ends up being the appropriate model in certain cases where there is more variability than would be predicted by the normal distribution. One of these cases is stock values, which have more variability (or “volatility,” to use the economic term) than would be predicted by the normal distribution.

There’s actually an entire family of t- distributions. They all have similar formulas (but the math is beyond the scope of this introductory course in statistics), and they all have slightly “fatter tails” than the normal distribution. But some are closer to normal than others.

The t- distributions that have higher “degrees of freedom” are closer to normal (degrees of freedom is a mathematical concept that we won’t study in this course, beyond merely mentioning it here). So, there’s a t- distribution “with one degree of freedom,” another t- distribution “with 2 degrees of freedom” which is slightly closer to normal, another t- distribution “with 3 degrees of freedom” which is a bit closer to normal than the previous ones, and so on.

The following picture illustrates this idea with just a couple of t- distributions (note that “degrees of freedom” is abbreviated “d.f.” on the picture):

The test statistic for our t-test for one population mean is a t -score which follows a t- distribution with (n – 1) degrees of freedom. Recall that each t- distribution is indexed according to “degrees of freedom.” Notice that, in the context of a test for a mean, the degrees of freedom depend on the sample size in the study.

Remember that we said that higher degrees of freedom indicate that the t- distribution is closer to normal. So in the context of a test for the mean, the larger the sample size , the higher the degrees of freedom, and the closer the t- distribution is to a normal z distribution .

As a result, in the context of a test for a mean, the effect of the t- distribution is most important for a study with a relatively small sample size .

We are now done introducing the t-distribution. What are implications of all of this?

• The null distribution of our t-test statistic is the t-distribution with (n-1) d.f. In other words, when Ho is true (i.e., when μ = μ 0 (mu = mu_zero)), our test statistic has a t-distribution with (n-1) d.f., and this is the distribution under which we find p-values.
• For a large sample size (n), the null distribution of the test statistic is approximately Z, so whether we use t(n – 1) or Z to calculate the p-values does not make a big difference.

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The Craft of Writing a Strong Hypothesis

Table of Contents

Writing a hypothesis is one of the essential elements of a scientific research paper. It needs to be to the point, clearly communicating what your research is trying to accomplish. A blurry, drawn-out, or complexly-structured hypothesis can confuse your readers. Or worse, the editor and peer reviewers.

A captivating hypothesis is not too intricate. This blog will take you through the process so that, by the end of it, you have a better idea of how to convey your research paper's intent in just one sentence.

What is a Hypothesis?

The first step in your scientific endeavor, a hypothesis, is a strong, concise statement that forms the basis of your research. It is not the same as a thesis statement , which is a brief summary of your research paper .

The sole purpose of a hypothesis is to predict your paper's findings, data, and conclusion. It comes from a place of curiosity and intuition . When you write a hypothesis, you're essentially making an educated guess based on scientific prejudices and evidence, which is further proven or disproven through the scientific method.

The reason for undertaking research is to observe a specific phenomenon. A hypothesis, therefore, lays out what the said phenomenon is. And it does so through two variables, an independent and dependent variable.

The independent variable is the cause behind the observation, while the dependent variable is the effect of the cause. A good example of this is “mixing red and blue forms purple.” In this hypothesis, mixing red and blue is the independent variable as you're combining the two colors at your own will. The formation of purple is the dependent variable as, in this case, it is conditional to the independent variable.

Different Types of Hypotheses‌

Types of hypotheses

Some would stand by the notion that there are only two types of hypotheses: a Null hypothesis and an Alternative hypothesis. While that may have some truth to it, it would be better to fully distinguish the most common forms as these terms come up so often, which might leave you out of context.

Apart from Null and Alternative, there are Complex, Simple, Directional, Non-Directional, Statistical, and Associative and casual hypotheses. They don't necessarily have to be exclusive, as one hypothesis can tick many boxes, but knowing the distinctions between them will make it easier for you to construct your own.

1. Null hypothesis

A null hypothesis proposes no relationship between two variables. Denoted by H 0 , it is a negative statement like “Attending physiotherapy sessions does not affect athletes' on-field performance.” Here, the author claims physiotherapy sessions have no effect on on-field performances. Even if there is, it's only a coincidence.

2. Alternative hypothesis

Considered to be the opposite of a null hypothesis, an alternative hypothesis is donated as H1 or Ha. It explicitly states that the dependent variable affects the independent variable. A good  alternative hypothesis example is “Attending physiotherapy sessions improves athletes' on-field performance.” or “Water evaporates at 100 °C. ” The alternative hypothesis further branches into directional and non-directional.

• Directional hypothesis: A hypothesis that states the result would be either positive or negative is called directional hypothesis. It accompanies H1 with either the ‘<' or ‘>' sign.
• Non-directional hypothesis: A non-directional hypothesis only claims an effect on the dependent variable. It does not clarify whether the result would be positive or negative. The sign for a non-directional hypothesis is ‘≠.'

3. Simple hypothesis

A simple hypothesis is a statement made to reflect the relation between exactly two variables. One independent and one dependent. Consider the example, “Smoking is a prominent cause of lung cancer." The dependent variable, lung cancer, is dependent on the independent variable, smoking.

4. Complex hypothesis

In contrast to a simple hypothesis, a complex hypothesis implies the relationship between multiple independent and dependent variables. For instance, “Individuals who eat more fruits tend to have higher immunity, lesser cholesterol, and high metabolism.” The independent variable is eating more fruits, while the dependent variables are higher immunity, lesser cholesterol, and high metabolism.

5. Associative and casual hypothesis

Associative and casual hypotheses don't exhibit how many variables there will be. They define the relationship between the variables. In an associative hypothesis, changing any one variable, dependent or independent, affects others. In a casual hypothesis, the independent variable directly affects the dependent.

6. Empirical hypothesis

Also referred to as the working hypothesis, an empirical hypothesis claims a theory's validation via experiments and observation. This way, the statement appears justifiable and different from a wild guess.

Say, the hypothesis is “Women who take iron tablets face a lesser risk of anemia than those who take vitamin B12.” This is an example of an empirical hypothesis where the researcher  the statement after assessing a group of women who take iron tablets and charting the findings.

7. Statistical hypothesis

The point of a statistical hypothesis is to test an already existing hypothesis by studying a population sample. Hypothesis like “44% of the Indian population belong in the age group of 22-27.” leverage evidence to prove or disprove a particular statement.

Characteristics of a Good Hypothesis

Writing a hypothesis is essential as it can make or break your research for you. That includes your chances of getting published in a journal. So when you're designing one, keep an eye out for these pointers:

• A research hypothesis has to be simple yet clear to look justifiable enough.
• It has to be testable — your research would be rendered pointless if too far-fetched into reality or limited by technology.
• It has to be precise about the results —what you are trying to do and achieve through it should come out in your hypothesis.
• A research hypothesis should be self-explanatory, leaving no doubt in the reader's mind.
• If you are developing a relational hypothesis, you need to include the variables and establish an appropriate relationship among them.
• A hypothesis must keep and reflect the scope for further investigations and experiments.

Separating a Hypothesis from a Prediction

Outside of academia, hypothesis and prediction are often used interchangeably. In research writing, this is not only confusing but also incorrect. And although a hypothesis and prediction are guesses at their core, there are many differences between them.

A hypothesis is an educated guess or even a testable prediction validated through research. It aims to analyze the gathered evidence and facts to define a relationship between variables and put forth a logical explanation behind the nature of events.

Predictions are assumptions or expected outcomes made without any backing evidence. They are more fictionally inclined regardless of where they originate from.

For this reason, a hypothesis holds much more weight than a prediction. It sticks to the scientific method rather than pure guesswork. "Planets revolve around the Sun." is an example of a hypothesis as it is previous knowledge and observed trends. Additionally, we can test it through the scientific method.

Whereas "COVID-19 will be eradicated by 2030." is a prediction. Even though it results from past trends, we can't prove or disprove it. So, the only way this gets validated is to wait and watch if COVID-19 cases end by 2030.

Finally, How to Write a Hypothesis

Quick tips on writing a hypothesis

1.  Be clear about your research question

A hypothesis should instantly address the research question or the problem statement. To do so, you need to ask a question. Understand the constraints of your undertaken research topic and then formulate a simple and topic-centric problem. Only after that can you develop a hypothesis and further test for evidence.

2. Carry out a recce

Once you have your research's foundation laid out, it would be best to conduct preliminary research. Go through previous theories, academic papers, data, and experiments before you start curating your research hypothesis. It will give you an idea of your hypothesis's viability or originality.

Making use of references from relevant research papers helps draft a good research hypothesis. SciSpace Discover offers a repository of over 270 million research papers to browse through and gain a deeper understanding of related studies on a particular topic. Additionally, you can use SciSpace Copilot , your AI research assistant, for reading any lengthy research paper and getting a more summarized context of it. A hypothesis can be formed after evaluating many such summarized research papers. Copilot also offers explanations for theories and equations, explains paper in simplified version, allows you to highlight any text in the paper or clip math equations and tables and provides a deeper, clear understanding of what is being said. This can improve the hypothesis by helping you identify potential research gaps.

3. Create a 3-dimensional hypothesis

Variables are an essential part of any reasonable hypothesis. So, identify your independent and dependent variable(s) and form a correlation between them. The ideal way to do this is to write the hypothetical assumption in the ‘if-then' form. If you use this form, make sure that you state the predefined relationship between the variables.

In another way, you can choose to present your hypothesis as a comparison between two variables. Here, you must specify the difference you expect to observe in the results.

4. Write the first draft

Now that everything is in place, it's time to write your hypothesis. For starters, create the first draft. In this version, write what you expect to find from your research.

Clearly separate your independent and dependent variables and the link between them. Don't fixate on syntax at this stage. The goal is to ensure your hypothesis addresses the issue.

5. Proof your hypothesis

After preparing the first draft of your hypothesis, you need to inspect it thoroughly. It should tick all the boxes, like being concise, straightforward, relevant, and accurate. Your final hypothesis has to be well-structured as well.

Research projects are an exciting and crucial part of being a scholar. And once you have your research question, you need a great hypothesis to begin conducting research. Thus, knowing how to write a hypothesis is very important.

Now that you have a firmer grasp on what a good hypothesis constitutes, the different kinds there are, and what process to follow, you will find it much easier to write your hypothesis, which ultimately helps your research.

Now it's easier than ever to streamline your research workflow with SciSpace Discover . Its integrated, comprehensive end-to-end platform for research allows scholars to easily discover, write and publish their research and fosters collaboration.

It includes everything you need, including a repository of over 270 million research papers across disciplines, SEO-optimized summaries and public profiles to show your expertise and experience.

If you found these tips on writing a research hypothesis useful, head over to our blog on Statistical Hypothesis Testing to learn about the top researchers, papers, and institutions in this domain.

Frequently Asked Questions (FAQs)

1. what is the definition of hypothesis.

According to the Oxford dictionary, a hypothesis is defined as “An idea or explanation of something that is based on a few known facts, but that has not yet been proved to be true or correct”.

2. What is an example of hypothesis?

The hypothesis is a statement that proposes a relationship between two or more variables. An example: "If we increase the number of new users who join our platform by 25%, then we will see an increase in revenue."

3. What is an example of null hypothesis?

A null hypothesis is a statement that there is no relationship between two variables. The null hypothesis is written as H0. The null hypothesis states that there is no effect. For example, if you're studying whether or not a particular type of exercise increases strength, your null hypothesis will be "there is no difference in strength between people who exercise and people who don't."

4. What are the types of research?

• Fundamental research

• Applied research

• Qualitative research

• Quantitative research

• Mixed research

• Exploratory research

• Longitudinal research

• Cross-sectional research

• Field research

• Laboratory research

• Fixed research

• Flexible research

• Action research

• Policy research

• Classification research

• Comparative research

• Causal research

• Inductive research

• Deductive research

5. How to write a hypothesis?

• Your hypothesis should be able to predict the relationship and outcome.

• Avoid wordiness by keeping it simple and brief.

• Your hypothesis should contain observable and testable outcomes.

• Your hypothesis should be relevant to the research question.

6. What are the 2 types of hypothesis?

• Null hypotheses are used to test the claim that "there is no difference between two groups of data".

• Alternative hypotheses test the claim that "there is a difference between two data groups".

7. Difference between research question and research hypothesis?

A research question is a broad, open-ended question you will try to answer through your research. A hypothesis is a statement based on prior research or theory that you expect to be true due to your study. Example - Research question: What are the factors that influence the adoption of the new technology? Research hypothesis: There is a positive relationship between age, education and income level with the adoption of the new technology.

8. What is plural for hypothesis?

The plural of hypothesis is hypotheses. Here's an example of how it would be used in a statement, "Numerous well-considered hypotheses are presented in this part, and they are supported by tables and figures that are well-illustrated."

9. What is the red queen hypothesis?

The red queen hypothesis in evolutionary biology states that species must constantly evolve to avoid extinction because if they don't, they will be outcompeted by other species that are evolving. Leigh Van Valen first proposed it in 1973; since then, it has been tested and substantiated many times.

10. Who is known as the father of null hypothesis?

The father of the null hypothesis is Sir Ronald Fisher. He published a paper in 1925 that introduced the concept of null hypothesis testing, and he was also the first to use the term itself.

11. When to reject null hypothesis?

You need to find a significant difference between your two populations to reject the null hypothesis. You can determine that by running statistical tests such as an independent sample t-test or a dependent sample t-test. You should reject the null hypothesis if the p-value is less than 0.05.

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How to write a research hypothesis

Last updated

19 January 2023

Reviewed by

Miroslav Damyanov

Start with a broad subject matter that excites you, so your curiosity will motivate your work. Conduct a literature search to determine the range of questions already addressed and spot any holes in the existing research.

Narrow the topics that interest you and determine your research question. Rather than focusing on a hole in the research, you might choose to challenge an existing assumption, a process called problematization. You may also find yourself with a short list of questions or related topics.

Use the FINER method to determine the single problem you'll address with your research. FINER stands for:

I nteresting

You need a feasible research question, meaning that there is a way to address the question. You should find it interesting, but so should a larger audience. Rather than repeating research that others have already conducted, your research hypothesis should test something novel or unique. 

The research must fall into accepted ethical parameters as defined by the government of your country and your university or college if you're an academic. You'll also need to come up with a relevant question since your research should provide a contribution to the existing research area.

This process typically narrows your shortlist down to a single problem you'd like to study and the variable you want to test. You're ready to write your hypothesis statements.

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• Types of research hypotheses

It is important to narrow your topic down to one idea before trying to write your research hypothesis. You'll only test one problem at a time. To do this, you'll write two hypotheses – a null hypothesis (H0) and an alternative hypothesis (Ha).

You'll come across many terms related to developing a research hypothesis or referring to a specific type of hypothesis. Let's take a quick look at these terms.

Null hypothesis

The term null hypothesis refers to a research hypothesis type that assumes no statistically significant relationship exists within a set of observations or data. It represents a claim that assumes that any observed relationship is due to chance. Represented as H0, the null represents the conjecture of the research.

Alternative hypothesis

The alternative hypothesis accompanies the null hypothesis. It states that the situation presented in the null hypothesis is false or untrue, and claims an observed effect in your test. This is typically denoted by Ha or H(n), where “n” stands for the number of alternative hypotheses. You can have more than one alternative hypothesis. 

Simple hypothesis

The term simple hypothesis refers to a hypothesis or theory that predicts the relationship between two variables - the independent (predictor) and the dependent (predicted). 

Complex hypothesis

The term complex hypothesis refers to a model – either quantitative (mathematical) or qualitative . A complex hypothesis states the surmised relationship between two or more potentially related variables.

Directional hypothesis

When creating a statistical hypothesis, the directional hypothesis (the null hypothesis) states an assumption regarding one parameter of a population. Some academics call this the “one-sided” hypothesis. The alternative hypothesis indicates whether the researcher tests for a positive or negative effect by including either the greater than (">") or less than ("<") sign.

Non-directional hypothesis

We refer to the alternative hypothesis in a statistical research question as a non-directional hypothesis. It includes the not equal ("≠") sign to show that the research tests whether or not an effect exists without specifying the effect's direction (positive or negative).

Associative hypothesis

The term associative hypothesis assumes a link between two variables but stops short of stating that one variable impacts the other. Academic statistical literature asserts in this sense that correlation does not imply causation. So, although the hypothesis notes the correlation between two variables – the independent and dependent - it does not predict how the two interact.

Logical hypothesis

Typically used in philosophy rather than science, researchers can't test a logical hypothesis because the technology or data set doesn't yet exist. A logical hypothesis uses logic as the basis of its assumptions. 

In some cases, a logical hypothesis can become an empirical hypothesis once technology provides an opportunity for testing. Until that time, the question remains too expensive or complex to address. Note that a logical hypothesis is not a statistical hypothesis.

Empirical hypothesis

When we consider the opposite of a logical hypothesis, we call this an empirical or working hypothesis. This type of hypothesis considers a scientifically measurable question. A researcher can consider and test an empirical hypothesis through replicable tests, observations, and measurements.

Statistical hypothesis

The term statistical hypothesis refers to a test of a theory that uses representative statistical models to test relationships between variables to draw conclusions regarding a large population. This requires an existing large data set, commonly referred to as big data, or implementing a survey to obtain original statistical information to form a data set for the study. 

Testing this type of hypothesis requires the use of random samples. Note that the null and alternative hypotheses are used in statistical hypothesis testing.

Causal hypothesis

The term causal hypothesis refers to a research hypothesis that tests a cause-and-effect relationship. A causal hypothesis is utilized when conducting experimental or quasi-experimental research.

Descriptive hypothesis

The term descriptive hypothesis refers to a research hypothesis used in non-experimental research, specifying an influence in the relationship between two variables.

• What makes an effective research hypothesis?

An effective research hypothesis offers a clearly defined, specific statement, using simple wording that contains no assumptions or generalizations, and that you can test. A well-written hypothesis should predict the tested relationship and its outcome. It contains zero ambiguity and offers results you can observe and test. 

The research hypothesis should address a question relevant to a research area. Overall, your research hypothesis needs the following essentials:

Hypothesis Essential #1: Specificity & Clarity

Hypothesis Essential #2: Testability (Provability)

• How to develop a good research hypothesis

In developing your hypothesis statements, you must pre-plan some of your statistical analysis. Once you decide on your problem to examine, determine three aspects:

the parameter you'll test

the test's direction (left-tailed, right-tailed, or non-directional)

the hypothesized parameter value

Any quantitative research includes a hypothesized parameter value of a mean, a proportion, or the difference between two proportions. Here's how to note each parameter:

Single mean (μ)

Paired means (μd)

Single proportion (p)

Difference between two independent means (μ1−μ2)

Difference between two proportions (p1−p2)

Simple linear regression slope (β)

Correlation (ρ)

Defining these parameters and determining whether you want to test the mean, proportion, or differences helps you determine the statistical tests you'll conduct to analyze your data. When writing your hypothesis, you only need to decide which parameter to test and in what overarching way.

The null research hypothesis must include everyday language, in a single sentence, stating the problem you want to solve. Write it as an if-then statement with defined variables. Write an alternative research hypothesis that states the opposite.

• What is the correct format for writing a hypothesis?

The following example shows the proper format and textual content of a hypothesis. It follows commonly accepted academic standards.

Null hypothesis (H0): High school students who participate in varsity sports as opposed to those who do not, fail to score higher on leadership tests than students who do not participate.

Alternative hypothesis (H1): High school students who play a varsity sport as opposed to those who do not participate in team athletics will score higher on leadership tests than students who do not participate in athletics.

The research question tests the correlation between varsity sports participation and leadership qualities expressed as a score on leadership tests. It compares the population of athletes to non-athletes.

• What are the five steps of a hypothesis?

Once you decide on the specific problem or question you want to address, you can write your research hypothesis. Use this five-step system to hone your null hypothesis and generate your alternative hypothesis.

Step 1 : Create your research question. This topic should interest and excite you; answering it provides relevant information to an industry or academic area.

Step 2 : Conduct a literature review to gather essential existing research.

Step 3 : Write a clear, strong, simply worded sentence that explains your test parameter, test direction, and hypothesized parameter.

Step 4 : Read it a few times. Have others read it and ask them what they think it means. Refine your statement accordingly until it becomes understandable to everyone. While not everyone can or will comprehend every research study conducted, any person from the general population should be able to read your hypothesis and alternative hypothesis and understand the essential question you want to answer.

Step 5 : Re-write your null hypothesis until it reads simply and understandably. Write your alternative hypothesis.

What is the Red Queen hypothesis?

Some hypotheses are well-known, such as the Red Queen hypothesis. Choose your wording carefully, since you could become like the famed scientist Dr. Leigh Van Valen. In 1973, Dr. Van Valen proposed the Red Queen hypothesis to describe coevolutionary activity, specifically reciprocal evolutionary effects between species to explain extinction rates in the fossil record. 

Essentially, Van Valen theorized that to survive, each species remains in a constant state of adaptation, evolution, and proliferation, and constantly competes for survival alongside other species doing the same. Only by doing this can a species avoid extinction. Van Valen took the hypothesis title from the Lewis Carroll book, "Through the Looking Glass," which contains a key character named the Red Queen who explains to Alice that for all of her running, she's merely running in place.

• Getting started with your research

In conclusion, once you write your null hypothesis (H0) and an alternative hypothesis (Ha), you’ve essentially authored the elevator pitch of your research. These two one-sentence statements describe your topic in simple, understandable terms that both professionals and laymen can understand. They provide the starting point of your research project.

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Video transcript

How to Write a Hypothesis: A Step-by-Step Guide

Introduction

An overview of the research hypothesis, different types of hypotheses, variables in a hypothesis, how to formulate an effective research hypothesis, designing a study around your hypothesis.

The scientific method can derive and test predictions as hypotheses. Empirical research can then provide support (or lack thereof) for the hypotheses. Even failure to find support for a hypothesis still represents a valuable contribution to scientific knowledge. Let's look more closely at the idea of the hypothesis and the role it plays in research.

As much as the term exists in everyday language, there is a detailed development that informs the word "hypothesis" when applied to research. A good research hypothesis is informed by prior research and guides research design and data analysis , so it is important to understand how a hypothesis is defined and understood by researchers.

What is the simple definition of a hypothesis?

A hypothesis is a testable prediction about an outcome between two or more variables . It functions as a navigational tool in the research process, directing what you aim to predict and how.

What is the hypothesis for in research?

In research, a hypothesis serves as the cornerstone for your empirical study. It not only lays out what you aim to investigate but also provides a structured approach for your data collection and analysis.

Essentially, it bridges the gap between the theoretical and the empirical, guiding your investigation throughout its course.

What is an example of a hypothesis?

If you are studying the relationship between physical exercise and mental health, a suitable hypothesis could be: "Regular physical exercise leads to improved mental well-being among adults."

This statement constitutes a specific and testable hypothesis that directly relates to the variables you are investigating.

What makes a good hypothesis?

A good hypothesis possesses several key characteristics. Firstly, it must be testable, allowing you to analyze data through empirical means, such as observation or experimentation, to assess if there is significant support for the hypothesis. Secondly, a hypothesis should be specific and unambiguous, giving a clear understanding of the expected relationship between variables. Lastly, it should be grounded in existing research or theoretical frameworks , ensuring its relevance and applicability.

Understanding the types of hypotheses can greatly enhance how you construct and work with hypotheses. While all hypotheses serve the essential function of guiding your study, there are varying purposes among the types of hypotheses. In addition, all hypotheses stand in contrast to the null hypothesis, or the assumption that there is no significant relationship between the variables .

Here, we explore various kinds of hypotheses to provide you with the tools needed to craft effective hypotheses for your specific research needs. Bear in mind that many of these hypothesis types may overlap with one another, and the specific type that is typically used will likely depend on the area of research and methodology you are following.

Null hypothesis

The null hypothesis is a statement that there is no effect or relationship between the variables being studied. In statistical terms, it serves as the default assumption that any observed differences are due to random chance.

For example, if you're studying the effect of a drug on blood pressure, the null hypothesis might state that the drug has no effect.

Alternative hypothesis

Contrary to the null hypothesis, the alternative hypothesis suggests that there is a significant relationship or effect between variables.

Using the drug example, the alternative hypothesis would posit that the drug does indeed affect blood pressure. This is what researchers aim to prove.

Simple hypothesis

A simple hypothesis makes a prediction about the relationship between two variables, and only two variables.

For example, "Increased study time results in better exam scores." Here, "study time" and "exam scores" are the only variables involved.

Complex hypothesis

A complex hypothesis, as the name suggests, involves more than two variables. For instance, "Increased study time and access to resources result in better exam scores." Here, "study time," "access to resources," and "exam scores" are all variables.

This hypothesis refers to multiple potential mediating variables. Other hypotheses could also include predictions about variables that moderate the relationship between the independent variable and dependent variable .

Directional hypothesis

A directional hypothesis specifies the direction of the expected relationship between variables. For example, "Eating more fruits and vegetables leads to a decrease in heart disease."

Here, the direction of heart disease is explicitly predicted to decrease, due to effects from eating more fruits and vegetables. All hypotheses typically specify the expected direction of the relationship between the independent and dependent variable, such that researchers can test if this prediction holds in their data analysis .

Statistical hypothesis

A statistical hypothesis is one that is testable through statistical methods, providing a numerical value that can be analyzed. This is commonly seen in quantitative research .

For example, "There is a statistically significant difference in test scores between students who study for one hour and those who study for two."

Empirical hypothesis

An empirical hypothesis is derived from observations and is tested through empirical methods, often through experimentation or survey data . Empirical hypotheses may also be assessed with statistical analyses.

For example, "Regular exercise is correlated with a lower incidence of depression," could be tested through surveys that measure exercise frequency and depression levels.

Causal hypothesis

A causal hypothesis proposes that one variable causes a change in another. This type of hypothesis is often tested through controlled experiments.

For example, "Smoking causes lung cancer," assumes a direct causal relationship.

Associative hypothesis

Unlike causal hypotheses, associative hypotheses suggest a relationship between variables but do not imply causation.

For instance, "People who smoke are more likely to get lung cancer," notes an association but doesn't claim that smoking causes lung cancer directly.

Relational hypothesis

A relational hypothesis explores the relationship between two or more variables but doesn't specify the nature of the relationship.

For example, "There is a relationship between diet and heart health," leaves the nature of the relationship (causal, associative, etc.) open to interpretation.

Logical hypothesis

A logical hypothesis is based on sound reasoning and logical principles. It's often used in theoretical research to explore abstract concepts, rather than being based on empirical data.

For example, "If all men are mortal and Socrates is a man, then Socrates is mortal," employs logical reasoning to make its point.

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In any research hypothesis, variables play a critical role. These are the elements or factors that the researcher manipulates, controls, or measures. Understanding variables is essential for crafting a clear, testable hypothesis and for the stages of research that follow, such as data collection and analysis.

In the realm of hypotheses, there are generally two types of variables to consider: independent and dependent. Independent variables are what you, as the researcher, manipulate or change in your study. It's considered the cause in the relationship you're investigating. For instance, in a study examining the impact of sleep duration on academic performance, the independent variable would be the amount of sleep participants get.

Conversely, the dependent variable is the outcome you measure to gauge the effect of your manipulation. It's the effect in the cause-and-effect relationship. The dependent variable thus refers to the main outcome of interest in your study. In the same sleep study example, the academic performance, perhaps measured by exam scores or GPA, would be the dependent variable.

Beyond these two primary types, you might also encounter control variables. These are variables that could potentially influence the outcome and are therefore kept constant to isolate the relationship between the independent and dependent variables . For example, in the sleep and academic performance study, control variables could include age, diet, or even the subject of study.

By clearly identifying and understanding the roles of these variables in your hypothesis, you set the stage for a methodologically sound research project. It helps you develop focused research questions, design appropriate experiments or observations, and carry out meaningful data analysis . It's a step that lays the groundwork for the success of your entire study.

Crafting a strong, testable hypothesis is crucial for the success of any research project. It sets the stage for everything from your study design to data collection and analysis . Below are some key considerations to keep in mind when formulating your hypothesis:

• Be specific : A vague hypothesis can lead to ambiguous results and interpretations . Clearly define your variables and the expected relationship between them.
• Ensure testability : A good hypothesis should be testable through empirical means, whether by observation , experimentation, or other forms of data analysis.
• Ground in literature : Before creating your hypothesis, consult existing research and theories. This not only helps you identify gaps in current knowledge but also gives you valuable context and credibility for crafting your hypothesis.
• Use simple language : While your hypothesis should be conceptually sound, it doesn't have to be complicated. Aim for clarity and simplicity in your wording.
• State direction, if applicable : If your hypothesis involves a directional outcome (e.g., "increase" or "decrease"), make sure to specify this. You also need to think about how you will measure whether or not the outcome moved in the direction you predicted.
• Keep it focused : One of the common pitfalls in hypothesis formulation is trying to answer too many questions at once. Keep your hypothesis focused on a specific issue or relationship.
• Account for control variables : Identify any variables that could potentially impact the outcome and consider how you will control for them in your study.
• Be ethical : Make sure your hypothesis and the methods for testing it comply with ethical standards , particularly if your research involves human or animal subjects.

Designing your study involves multiple key phases that help ensure the rigor and validity of your research. Here we discuss these crucial components in more detail.

Literature review

Starting with a comprehensive literature review is essential. This step allows you to understand the existing body of knowledge related to your hypothesis and helps you identify gaps that your research could fill. Your research should aim to contribute some novel understanding to existing literature, and your hypotheses can reflect this. A literature review also provides valuable insights into how similar research projects were executed, thereby helping you fine-tune your own approach.

Research methods

Choosing the right research methods is critical. Whether it's a survey, an experiment, or observational study, the methodology should be the most appropriate for testing your hypothesis. Your choice of methods will also depend on whether your research is quantitative, qualitative, or mixed-methods. Make sure the chosen methods align well with the variables you are studying and the type of data you need.

Preliminary research

Before diving into a full-scale study, it’s often beneficial to conduct preliminary research or a pilot study . This allows you to test your research methods on a smaller scale, refine your tools, and identify any potential issues. For instance, a pilot survey can help you determine if your questions are clear and if the survey effectively captures the data you need. This step can save you both time and resources in the long run.

Data analysis

Finally, planning your data analysis in advance is crucial for a successful study. Decide which statistical or analytical tools are most suited for your data type and research questions . For quantitative research, you might opt for t-tests, ANOVA, or regression analyses. For qualitative research , thematic analysis or grounded theory may be more appropriate. This phase is integral for interpreting your results and drawing meaningful conclusions in relation to your research question.

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