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Hypothesis Testing with One Sample

Null and Alternative Hypotheses

OpenStaxCollege

[latexpage]

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: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

H a : The alternative hypothesis: It is 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 (including one of the co-authors in research work) 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.

H 0 : No more than 30% 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%. State the null and alternative hypotheses.

H 0 : The drug reduces cholesterol by 25%. p = 0.25

H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25

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:

H 0 : 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

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

H 0 : 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

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

H 0 : p ≤ 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% 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

<!– ??? –>

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.

Chapter Review

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:

Formula Review

H 0 and H a are contradictory.

If α ≤ p -value, then do not reject H 0 .

If α > p -value, then reject H 0 .

α is preconceived. Its value is set before the hypothesis test starts. The p -value is calculated from the data.

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. What is the random variable? Describe in words.

The random variable is the mean Internet speed in Megabits per second.

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. State the null and alternative hypotheses.

The American family has an average of two children. What is the random variable? Describe in words.

The random variable is the mean number of children an American family has.

The mean entry level salary of an employee at a company is 💲58,000. You believe it is higher for IT professionals in the company. State the null and alternative hypotheses.

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the proportion is actually less. What is the random variable? Describe in words.

The random variable is the proportion of people picked at random in Times Square visiting the city.

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the claim is correct. State the null and alternative hypotheses.

In a population of fish, approximately 42% are female. A test is conducted to see if, in fact, the proportion is less. State the null and alternative hypotheses.

Suppose that a recent article stated that the mean time spent in jail by a first–time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was 3 years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. If you were conducting a hypothesis test to determine if the mean length of jail time has increased, what would the null and alternative hypotheses be? The distribution of the population is normal.

A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be 15 years, what would the null and alternative hypotheses be?

• H 0 : __________
• H a : __________
• H 0 : μ = 15
• H a : μ ≠ 15

The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. If you were conducting a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population, what would the null and alternative hypotheses be?

Some of the following statements refer to the null hypothesis, some to the alternate hypothesis.

State the null hypothesis, H 0 , and the alternative hypothesis. H a , in terms of the appropriate parameter ( μ or p ).

• The mean number of years Americans work before retiring is 34.
• At most 60% of Americans vote in presidential elections.
• The mean starting salary for San Jose State University graduates is at least 💲100,000 per year.
• Twenty-nine percent of high school seniors get drunk each month.
• Fewer than 5% of adults ride the bus to work in Los Angeles.
• The mean number of cars a person owns in her lifetime is not more than ten.
• About half of Americans prefer to live away from cities, given the choice.
• Europeans have a mean paid vacation each year of six weeks.
• The chance of developing breast cancer is under 11% for women.
• Private universities’ mean tuition cost is more than 💲20,000 per year.
• H 0 : μ = 34; H a : μ ≠ 34
• H 0 : p ≤ 0.60; H a : p > 0.60
• H 0 : μ ≥ 100,000; H a : μ < 100,000
• H 0 : p = 0.29; H a : p ≠ 0.29
• H 0 : p = 0.05; H a : p < 0.05
• H 0 : μ ≤ 10; H a : μ > 10
• H 0 : p = 0.50; H a : p ≠ 0.50
• H 0 : μ = 6; H a : μ ≠ 6
• H 0 : p ≥ 0.11; H a : p < 0.11
• H 0 : μ ≤ 20,000; H a : μ > 20,000

Over the past few decades, public health officials have examined the link between weight concerns and teen girls’ smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than thirty percent of the teen girls smoke to stay thin? The alternative hypothesis is:

• p < 0.30
• p > 0.30

A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Harry Potter movie. She surveys 84 of her students and finds that 11 attended the midnight showing. An appropriate alternative hypothesis is:

• p > 0.20
• p < 0.20

Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. The null and alternative hypotheses are:

• H o : \(\overline{x}\) = 4.5, H a : \(\overline{x}\) > 4.5
• H o : μ ≥ 4.5, H a : μ < 4.5
• H o : μ = 4.75, H a : μ > 4.75
• H o : μ = 4.5, H a : μ > 4.5

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm.

Null and Alternative Hypotheses Copyright © 2013 by OpenStaxCollege is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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3.1: The Fundamentals of Hypothesis Testing

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• Page ID 2883

• Diane Kiernan
• SUNY College of Environmental Science and Forestry via OpenSUNY

The previous two chapters introduced methods for organizing and summarizing sample data, and using sample statistics to estimate population parameters. This chapter introduces the next major topic of inferential statistics: hypothesis testing.

A hypothesis is a statement or claim about a property of a population.

The Fundamentals of Hypothesis Testing

When conducting scientific research, typically there is some known information, perhaps from some past work or from a long accepted idea. We want to test whether this claim is believable. This is the basic idea behind a hypothesis test:

• State what we think is true.
• Quantify how confident we are about our claim.
• Use sample statistics to make inferences about population parameters.

For example, past research tells us that the average life span for a hummingbird is about four years. You have been studying the hummingbirds in the southeastern United States and find a sample mean lifespan of 4.8 years. Should you reject the known or accepted information in favor of your results? How confident are you in your estimate? At what point would you say that there is enough evidence to reject the known information and support your alternative claim? How far from the known mean of four years can the sample mean be before we reject the idea that the average lifespan of a hummingbird is four years?

Definition: hypothesis testing

Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of a population.

A hypothesis is a claim or statement about a characteristic of a population of interest to us. A hypothesis test is a way for us to use our sample statistics to test a specific claim.

Example \(\PageIndex{1}\):

The population mean weight is known to be 157 lb. We want to test the claim that the mean weight has increased.

Example \(\PageIndex{2}\):

Two years ago, the proportion of infected plants was 37%. We believe that a treatment has helped, and we want to test the claim that there has been a reduction in the proportion of infected plants.

Components of a Formal Hypothesis Test

The null hypothesis is a statement about the value of a population parameter, such as the population mean (µ) or the population proportion ( p ). It contains the condition of equality and is denoted as H 0 (H-naught).

H 0 : µ = 157 or H0 : p = 0.37

The alternative hypothesis is the claim to be tested, the opposite of the null hypothesis. It contains the value of the parameter that we consider plausible and is denoted as H 1 .

H 1 : µ > 157 or H1 : p ≠ 0.37

The test statistic is a value computed from the sample data that is used in making a decision about the rejection of the null hypothesis. The test statistic converts the sample mean ( x̄ ) or sample proportion ( p̂ ) to a Z- or t-score under the assumption that the null hypothesis is true. It is used to decide whether the difference between the sample statistic and the hypothesized claim is significant.

The p-value is the area under the curve to the left or right of the test statistic. It is compared to the level of significance (α).

The critical value is the value that defines the rejection zone (the test statistic values that would lead to rejection of the null hypothesis). It is defined by the level of significance.

The level of significance (α) is the probability that the test statistic will fall into the critical region when the null hypothesis is true. This level is set by the researcher.

The conclusion is the final decision of the hypothesis test. The conclusion must always be clearly stated, communicating the decision based on the components of the test. It is important to realize that we never prove or accept the null hypothesis. We are merely saying that the sample evidence is not strong enough to warrant the rejection of the null hypothesis. The conclusion is made up of two parts:

1) Reject or fail to reject the null hypothesis, and 2) there is or is not enough evidence to support the alternative claim.

Option 1) Reject the null hypothesis (H0). This means that you have enough statistical evidence to support the alternative claim (H 1 ).

Option 2) Fail to reject the null hypothesis (H0). This means that you do NOT have enough evidence to support the alternative claim (H 1 ).

Another way to think about hypothesis testing is to compare it to the US justice system. A defendant is innocent until proven guilty (Null hypothesis—innocent). The prosecuting attorney tries to prove that the defendant is guilty (Alternative hypothesis—guilty). There are two possible conclusions that the jury can reach. First, the defendant is guilty (Reject the null hypothesis). Second, the defendant is not guilty (Fail to reject the null hypothesis). This is NOT the same thing as saying the defendant is innocent! In the first case, the prosecutor had enough evidence to reject the null hypothesis (innocent) and support the alternative claim (guilty). In the second case, the prosecutor did NOT have enough evidence to reject the null hypothesis (innocent) and support the alternative claim of guilty.

The Null and Alternative Hypotheses

There are three different pairs of null and alternative hypotheses:

Table \(PageIndex{1}\): The rejection zone for a two-sided hypothesis test.

where c is some known value.

A Two-sided Test

This tests whether the population parameter is equal to, versus not equal to, some specific value.

Ho: μ = 12 vs. H 1 : μ ≠ 12

The critical region is divided equally into the two tails and the critical values are ± values that define the rejection zones.

Example \(\PageIndex{3}\):

A forester studying diameter growth of red pine believes that the mean diameter growth will be different if a fertilization treatment is applied to the stand.

• Ho: μ = 1.2 in./ year
• H 1 : μ ≠ 1.2 in./ year

This is a two-sided question, as the forester doesn’t state whether population mean diameter growth will increase or decrease.

A Right-sided Test

This tests whether the population parameter is equal to, versus greater than, some specific value.

Ho: μ = 12 vs. H 1 : μ > 12

The critical region is in the right tail and the critical value is a positive value that defines the rejection zone.

Example \(\PageIndex{4}\):

A biologist believes that there has been an increase in the mean number of lakes infected with milfoil, an invasive species, since the last study five years ago.

• Ho: μ = 15 lakes
• H1: μ >15 lakes

This is a right-sided question, as the biologist believes that there has been an increase in population mean number of infected lakes.

A Left-sided Test

This tests whether the population parameter is equal to, versus less than, some specific value.

Ho: μ = 12 vs. H 1 : μ < 12

The critical region is in the left tail and the critical value is a negative value that defines the rejection zone.

Example \(\PageIndex{5}\):

A scientist’s research indicates that there has been a change in the proportion of people who support certain environmental policies. He wants to test the claim that there has been a reduction in the proportion of people who support these policies.

• Ho: p = 0.57
• H 1 : p < 0.57

This is a left-sided question, as the scientist believes that there has been a reduction in the true population proportion.

Statistically Significant

When the observed results (the sample statistics) are unlikely (a low probability) under the assumption that the null hypothesis is true, we say that the result is statistically significant, and we reject the null hypothesis. This result depends on the level of significance, the sample statistic, sample size, and whether it is a one- or two-sided alternative hypothesis.

Types of Errors

When testing, we arrive at a conclusion of rejecting the null hypothesis or failing to reject the null hypothesis. Such conclusions are sometimes correct and sometimes incorrect (even when we have followed all the correct procedures). We use incomplete sample data to reach a conclusion and there is always the possibility of reaching the wrong conclusion. There are four possible conclusions to reach from hypothesis testing. Of the four possible outcomes, two are correct and two are NOT correct.

Table \(\PageIndex{2}\). Possible outcomes from a hypothesis test.

A Type I error is when we reject the null hypothesis when it is true. The symbol α (alpha) is used to represent Type I errors. This is the same alpha we use as the level of significance. By setting alpha as low as reasonably possible, we try to control the Type I error through the level of significance.

A Type II error is when we fail to reject the null hypothesis when it is false. The symbol β(beta) is used to represent Type II errors.

In general, Type I errors are considered more serious. One step in the hypothesis test procedure involves selecting the significance level ( α ), which is the probability of rejecting the null hypothesis when it is correct. So the researcher can select the level of significance that minimizes Type I errors. However, there is a mathematical relationship between α, β, and n (sample size).

• As α increases, β decreases
• As α decreases, β increases
• As sample size increases (n), both α and β decrease

The natural inclination is to select the smallest possible value for α, thinking to minimize the possibility of causing a Type I error. Unfortunately, this forces an increase in Type II errors. By making the rejection zone too small, you may fail to reject the null hypothesis, when, in fact, it is false. Typically, we select the best sample size and level of significance, automatically setting β.

Power of the Test

A Type II error (β) is the probability of failing to reject a false null hypothesis. It follows that 1-β is the probability of rejecting a false null hypothesis. This probability is identified as the power of the test, and is often used to gauge the test’s effectiveness in recognizing that a null hypothesis is false.

Definition: power of the test

The probability that at a fixed level α significance test will reject H0, when a particular alternative value of the parameter is true is called the power of the test.

Power is also directly linked to sample size. For example, suppose the null hypothesis is that the mean fish weight is 8.7 lb. Given sample data, a level of significance of 5%, and an alternative weight of 9.2 lb., we can compute the power of the test to reject μ = 8.7 lb. If we have a small sample size, the power will be low. However, increasing the sample size will increase the power of the test. Increasing the level of significance will also increase power. A 5% test of significance will have a greater chance of rejecting the null hypothesis than a 1% test because the strength of evidence required for the rejection is less. Decreasing the standard deviation has the same effect as increasing the sample size: there is more information about μ.

How to Write a Null and Alternative Hypothesis: A Guide with Examples

11 December 2023

last updated

When undertaking a qualitative or quantitative research project, researchers must first formulate a research question, from which they develop a hypothesis. By definition, a hypothesis is a prediction that a researcher makes about the research question and can either be affirmative or negative. In this case, a research question has three main components: variables (independent and dependent), a population sample, and the relation between the variables. When the prediction contradicts the research question, it is referred to as a null hypothesis. In short, a null hypothesis is a statement that implies there is no relationship between independent and dependent variables. Hence, researchers need to learn how to write a good null and alternative hypothesis to present quality studies.

General Aspect of Writing a Null and Alternative Hypothesis

Students with qualitative or quantitative research assignments must learn how to formulate and write a good research question and hypothesis. By definition, a hypothesis is an assumption or prediction that a researcher makes before undertaking an experimental investigation. Basically, academic standards require such a prediction to be a precise and testable statement, meaning that researchers must prove or disapprove of it in the course of the assignment. In this case, the main components of a hypothesis are variables (independent and dependent), a population sample, and the relation between the variables. Therefore, a research hypothesis is a prediction that researchers write about the relationship between two or more variables. In turn, the research inquiry is the process that seeks to answer the research question and, in the process, test the hypothesis by confirming or disapproving it.

Types of Hypotheses

There are several types of hypotheses, including an alternative hypothesis, a null hypothesis, a directional hypothesis, and a non-directional hypothesis. Basically, the directional hypothesis is a prediction of how the independent variable affects the dependent variable. In contrast, the non-directional hypothesis predicts that the independent variable influences the dependent variable, but does not specify how. Regardless of the type, all hypotheses are about predicting the relationship between the independent and dependent variables.

What Is a Null and Alternative Hypothesis

A null hypothesis, usually symbolized as “H0,” is a statement that contradicts the research hypothesis. In other words, it is a negative statement, indicating that there is no relationship between the independent and dependent variables. By testing the null hypothesis, a researcher can determine whether the inquiry results are due to the chance or the effect of manipulating the dependent variable. In most instances, a null hypothesis corresponds with an alternative hypothesis, a positive statement that covers a relationship that exists between the independent and dependent variables. Also, it is highly recommendable that a researcher should write the alternative hypothesis first before the null hypothesis.

10 Examples of Research Questions with H0 and H1 Hypotheses

Before developing a hypothesis, a researcher must formulate the research question. Then, the next step is to transform the question into a negative statement that claims the lack of a relationship between the independent and dependent variables. Alternatively, researchers can change the question into a positive statement that includes a relationship that exists between the variables. In turn, this latter statement becomes the alternative hypothesis and is symbolized as H1. Hence, some of the examples of research questions and hull and alternative hypotheses are as follows:

1. Do physical exercises help individuals to age gracefully?

A Null Hypothesis (H0): Physical exercises are not a guarantee for graceful old age.

An Alternative Hypothesis (H1): Engaging in physical exercises enables individuals to remain healthy and active into old age.

2. What are the implications of therapeutic interventions in the fight against substance abuse?

H0: Therapeutic interventions are of no help in the fight against substance abuse.

H1: Exposing individuals with substance abuse disorders to therapeutic interventions help control and even stop their addictions.

3. How do sexual orientation and gender identity affect the experiences of late adolescents in foster care?

H0: Sexual orientation and gender identity have no effects on the experiences of late adolescents in foster care.

H1: The reality of stereotypes in society makes sexual orientation and gender identity factors complicate the experiences of late adolescents in foster care.

4. Does income inequality contribute to crime in high-density urban areas?

H0: There is no correlation between income inequality and incidences of crime in high-density urban areas.

H1: The high crime rates in high-density urban areas are due to the incidence of income inequality in those areas.

5. Does placement in foster care impact individuals’ mental health?

H0: There is no correlation between being in foster care and having mental health problems.

H1: Individuals placed in foster care experience anxiety and depression at one point in their life.

6. Do assistive devices and technologies lessen the mobility challenges of older adults with a stroke?

H0: Assistive devices and technologies do not provide any assistance to the mobility of older adults diagnosed with a stroke.

H1: Assistive devices and technologies enhance the mobility of older adults diagnosed with a stroke.

7. Does race identity undermine classroom participation?

H0: There is no correlation between racial identity and the ability to participate in classroom learning.

H1: Students from racial minorities are not as active as white students in classroom participation.

8. Do high school grades determine future success?

H0: There is no correlation between how one performs in high school and their success level in life.

H1: Attaining high grades in high school positions one for greater success in the future personal and professional lives.

9. Does critical thinking predict academic achievement?

H0: There is no correlation between critical thinking and academic achievement.

H1: Being a critical thinker is a pathway to academic success.

10. What benefits does group therapy provide to victims of domestic violence?

H0: Group therapy does not help victims of domestic violence because individuals prefer to hide rather than expose their shame.

H1: Group therapy provides domestic violence victims with a platform to share their hurt and connect with others with similar experiences.

Summing Up on How to Write a Null and Alternative Hypothesis

The formulation of research questions in qualitative and quantitative assignments helps students develop a hypothesis for their experiment. In this case, learning how to write a good hypothesis that helps students and researchers to make their research relevant. Basically, the difference between a null and alternative hypothesis is that the former contradicts the research question, while the latter affirms it. In short, a null hypothesis is a negative statement relative to the research question, and an alternative hypothesis is a positive statement. Moreover, it is important to note that developing the null hypothesis at the beginning of the assignment is for prediction purposes. As such, the research work answers the research question and confirms or disapproves of the hypothesis. Hence, some of the tips that students and researchers need to know when developing a null hypothesis include:

• Formulate a research question that specifies the relationship between an independent variable and a dependent variable.
• Develop an alternative hypothesis that says a relationship that exists between the variables.
• Develop a null hypothesis that says a relationship that does not exist between the variables.
• Conduct the research to answer the research question, which allows the confirmation of a disapproval of a null hypothesis.

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Exploring the Null Hypothesis: Definition and Purpose

Updated: July 5, 2023 by Ken Feldman

Hypothesis testing is a branch of statistics in which, using data from a sample, an inference is made about a population parameter or a population probability distribution .

First, a hypothesis statement and assumption is made about the population parameter or probability distribution. This initial statement is called the Null Hypothesis and is denoted by H o. An alternative or alternate hypothesis (denoted Ha ), is then stated which will be the opposite of the Null Hypothesis.

The hypothesis testing process and analysis involves using sample data to determine whether or not you can be statistically confident that you can reject or fail to reject the H o. If the H o is rejected, the statistical conclusion is that the alternative or alternate hypothesis Ha is true.

Overview: What is the Null Hypothesis (Ho)? 

Hypothesis testing applies to all forms of statistical inquiry. For example, it can be used to determine whether there are differences between population parameters or an understanding about slopes of regression lines or equality of probability distributions.

In all cases, the first thing you do is state the Null and Alternate Hypotheses. The word Null in the context of hypothesis testing means “nothing” or “zero.”  

As an example, if we wanted to test whether there was a difference in two population means based on the calculations from two samples, we would state the Null Hypothesis in the form of: 

Ho: mu1 = mu2 or mu1- mu2 = 0  

In other words, there is no difference, or the difference is zero. Note that the notation is in the form of a population parameter, not a sample statistic. 

Since you are using sample data to make your inferences about the population, it’s possible you’ll make an error. In the case of the Null Hypothesis, we can make one of two errors.

•   Type 1 , or alpha error: An alpha error is when you mistakenly reject the Null and believe that something significant happened. In other words, you believe that the means of the two populations are different when they aren’t.
• Type 2, or beta error: A beta error is when you fail to reject the null when you should have.  In this case, you missed something significant and failed to take action. 

A classic example is when you get the results back from your doctor after taking a blood test. If the doctor says you have an infection when you really don’t, that is an alpha error. That is thinking that there is something significant going on when there isn’t. We also call that a false positive. The doctor rejected the null that “there was zero infection” and missed the call.

On the other hand, if the doctor told you that everything was OK when you really did have an infection, then he made a beta, or type 2, error. He failed to reject the Null Hypothesis when he should have. That is called a false negative.

The decision to reject or not to reject the Null Hypothesis is based on three numbers. 

• Alpha, which you get to choose. Alpha is the risk you are willing to assume of falsely rejecting the Null. The typical values for alpha are 1%, 5%, or 10%. Depending on the importance of the conclusion, you only want to falsely claim a difference when there is none, 1%, 5%, or 10% of the time.
• Beta, which is typically 20%. This means you’re willing to be wrong 20% of the time in failing to reject the null when you should have. 
• P-value, which is calculated from the data. The p-value is the actual risk you have in being wrong if you reject the null. You would like that to be low.  

Your decision as to what to do about the null is made by comparing the alpha value (your assumed risk) with the p-value (actual risk). If the actual risk is lower than your assumed risk, you can feel comfortable in rejecting the null and claiming something has happened. But, if the actual risk is higher than your assumed risk you will be taking a bigger risk than you want by rejecting the null.

RELATED: NULL VS. ALTERNATIVE HYPOTHESIS

3 benefits of the null hypothesis .

The stating and testing of the null hypothesis is the foundation of hypothesis testing. By doing so, you set the parameters for your statistical inference.

1. Statistical assurance of determining differences between population parameters

Just looking at the mathematical difference between the means of two samples and making a decision is woefully inadequate. By statistically testing the null hypothesis, you will have more confidence in any inferences you want to make about populations based on your samples.

2. Statistically based estimation of the probability of a population distribution

Many statistical tests require assumptions of specific distributions. Many of these tests assume that the population follows the normal distribution . If it doesn’t, the test may be invalid.  

3. Assess the strength of your conclusions as to what to do with the null hypothesis

Hypothesis testing calculations will provide some relative strength to your decisions as to whether you reject or fail to reject the null hypothesis.

Why is the Null Hypothesis important to understand?

The interpretation of the statistics relative to the null hypothesis is what’s important.

1. Properly write the null hypothesis to properly capture what you are seeking to prove

The null is always written in the same format. That is, the lack of difference or some other condition. The alternative hypothesis can be written in three formats depending on what you want to prove. 

2. Frame your statement and select an appropriate alpha risk

You don’t want to place too big of a hurdle or burden on your decision-making relative to action on the null hypothesis by selecting an alpha value that is too high or too low.

3. There are decision errors when deciding on how to respond to the Null Hypothesis

Since your decision relative to rejecting or not rejecting the null is based on statistical calculations, it is important to understand how that decision works. 

An industry example of using the Null Hypothesis 

The new director of marketing just completed the rollout of a new marketing campaign targeting the Hispanic market. Early indications showed that the campaign was successful in increasing sales in the Hispanic market. 

He came to that conclusion by comparing a sample of sales prior to the campaign and current sales after implementation of the campaign. He was anxious to proudly tell his boss how successful the campaign was. But, he decided to first check with his Lean Six Sigma Black Belt to see whether she agreed with his conclusion.

The Black Belt first asked the director his tolerance for risk of being wrong by telling the boss the campaign was successful when in fact, it wasn’t. That was the alpha value. The Director picked 5% since he was new and didn’t want to make a false claim so early in his career. He also picked 20% as his beta value.  

When the Black Belt was done analyzing the data, she found out that the p-value was 15%.  That meant if the director told the VP the campaign worked, there was a 15% chance he would be wrong and that the campaign probably needed some revising. Since he was only willing to be wrong 5% of the time, the decision was to not reject the null since his 5% assumed risk was less than the 15% actual risk.

3 best practices when thinking about the Null Hypothesis 

Using hypothesis testing to help make better data-driven decisions requires that you properly address the Null Hypothesis. 

1. Always use the proper nomenclature when stating the Null Hypothesis 

The null will always be in the form of decisions regarding the population, not the sample. 

2. The Null Hypothesis will always be written as the absence of some parameter or process characteristic

The writing of the Alternate Hypothesis can vary, so be sure you understand exactly what condition you are testing against. 

3. Pick a reasonable alpha risk so you’re not always failing to reject the Null Hypothesis

Being too cautious will lead you to make beta errors, and you’ll never learn anything about your population data. 

Frequently Asked Questions (FAQ) about the Null Hypothesis

What form should the null hypothesis be written in.

The Null Hypothesis should always be in the form of no difference or zero and always refer to the state of the population, not the sample. 

What is an alpha error? 

An alpha error, or Type 1 error, is rejecting the Null Hypothesis and claiming a significant event has occurred when, in fact, that is not true and the Null should not have been rejected.

How do I use the alpha error and p-value to decide on what decision I should make about the Null Hypothesis? 

The most common way of answering this is, “If the p-value is low (less than the alpha), the Null should be rejected. If the p-value is high (greater than the alpha) then the Null should not be rejected.”

Becoming familiar with the Null Hypothesis (Ho)

The proper writing of the Null Hypothesis is the basis for applying hypothesis testing to help you make better data-driven decisions. The format of the Null will always be in the form of zero, or the non-existence of some condition. It will always refer to a population parameter and not the sample you use to do your hypothesis testing calculations.

Be aware of the two types of errors you can make when deciding on what to do with the Null. Select reasonable risks values for your alpha and beta risks. By comparing your alpha risk with the calculated risk computed from the data, you will have sufficient information to make a wise decision as to whether you should reject the Null Hypothesis or not.

About the Author

Ken Feldman

• Abnormal Psychology
• Assessment (IB)
• Biological Psychology
• Cognitive Psychology
• Criminology
• Developmental Psychology
• Extended Essay
• General Interest
• Health Psychology
• Human Relationships
• IB Psychology
• IB Psychology HL Extensions
• Internal Assessment (IB)
• Love and Marriage
• Post-Traumatic Stress Disorder
• Prejudice and Discrimination
• Qualitative Research Methods
• Research Methodology
• Revision and Exam Preparation
• Social and Cultural Psychology
• Studies and Theories
• Teaching Ideas

Travis Dixon October 24, 2016 Assessment (IB) , Internal Assessment (IB) , Research Methodology

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Updated June 2020

Writing good hypotheses in IB Psychology IAs is something many students find challenging. After moderating another 175+ IA’s this year I could see some common errors students were making. This post hopes to give a clear explanation with examples to help with this tricky task. 

Null and Alternative Hypotheses

Null hypothesis (h0).

Our teacher support pack has everything students and teachers need to get top marks in the IA. Download a Free preview from https://store.themantic-education.com/

The term “null” means having no value, significance or effect. It also refers to something associated with zero. A null hypothesis in a student’s IA, therefore, should state that there is (or will be) no effect of the IV on the DV. This is what we assume to be true until we have the evidence to suggest otherwise.

A common misconception is that the hypothesis is based on the sample in the study. Our hypotheses should actually be about the population from which we’ve drawn the sample, not the sample itself. Therefore, when writing our hypotheses we can use present tense instead of future tense (e.g. There is instead of There will be… ).

Having said that, in the IB Psych’ IA, the IB is apparently assuming the hypotheses are based on the sample (because variables need to be operationalized) so writing your hypotheses as predictions of what might happen in the experiment is fine (see below for examples).

IB Psych IA Tip: It’s fine (and even recommended) to state in your null hypotheses that there will be no significant difference between the two conditions in your experiment or any differences are due to chance (see footnote 1)

The Alternative Hypothesis (H1)

This is also referred to as the research hypothesis or the experimental hypothesis. It’s an alternative hypothesis to the null because if the null is not true, there must be an alternative explanation.

Generally speaking it’s not a prediction of what will happen in the study, but it’s an assumption about what is true for the population being studied. But, similar to the null hypothesis in the IB Psych IA you can (and should) write this about a prediction of what you think will happen in your study (see examples below).

This must be operationalized: it must be evident how the variables will be quantified, and may be either one- or two-tailed (directional or non-directional).

Read more: 

Operational Definitions

• Key Studies for the IA
• Lesson Idea: Inferential Statistics

To avoid issues with copying and plagiarism, the following examples are from studies that students cannot do for the internal assessment. Some are taken from this post on how to operationalize definitions of variables .

A Fictional Drug Trial

• H1: Taking Paroxetine  will decrease symptoms of PTSD.
• Ho: Taking paroxetine will not decrease symptoms of PTSD.

Operationalized (as if for an IB Psych IA):

• H1: The experimental group who take 20mg of Paroxetine (as a pill) every morning for 7 days will have a larger decrease in symptoms (as measured by the CAPs scale) when compared to the control group who will take an identical placebo pill every morning for 7 days.

A Fictional Study on Body Image*

• H1: Viewing media that portrays the thin ideal increases feelings of body image dissatisfaction.
• Ho: Types of media viewed does not affect body image dissatisfaction.
• H1: Watching a video portraying the thin ideal in a  Baywatch  film trailer will result in higher scores on the Body Shape Questionnaire (BSQ-34) compared with watching media with “normal” body types in the Grownups film trailer.

*This entire IA exemplar is included in the IA Teacher Support Pack.  

A fictional study on weight training.

• H1: Listening to music affects training performance.
• Ho: Music has no effect on training performance.
• H1:  Listening to heavy metal rock music (AC/DC songs) causes a difference in the number of push-ups performed compared to listening to classical music (Mozart’s symphony #41).

One vs. Two Tailed

It is important to know if your hypothesis is one or two-tailed. This will influence the type of inferential statistics test you use later. If you have a one-tailed hypotheses, you should use a one-tailed test. And if you have a two-tailed hypothesis? You guessed it – a two-tailed test.

The one vs two tailed debate still continues in Psychology ( read more ). The IB ignores this and makes it simple: one tailed hypotheses = one tailed test. No ifs, ands, or buts!

If you are predicting that one of your conditions in your experiment will have a higher value than the other, it’s one-tailed (because you know the direction of the effect – the IV is increasing the DV). Similarly, your hypothesis is one-tailed if you are predicting that manipulating the IV will cause a decrease in the DV.

However, if you think your IV will have an effect, but you’re not sure if it will increase  or  decrease it, this is two-tailed.

Of the three examples above, can you tell which one is two-tailed and which one is one-tailed?

Read more about operationally defining your variables in your hypotheses in this blog post .

Points to Remember

• Hypotheses are based on the population, not the sample, so you can write in present tense. However, the norm for IB Psych IA’s is to write in the future tense as a prediction of what will happen in your experiment.
• In IB IA’s, we’re hypothesizing about a causal relationship of an IV on a DV in a population – the hypotheses should reflect that causal relationship.
• Inferential tests are test of the null hypothesis (hence it’s called null hypothesis testing). We are conducting the tests to see the chances of obtaining our results even if the null is true (i.e. there is no effect).

Footnote 1: Saying “that there will be no significant difference between the two conditions in or any differences are due to chance” is technically an incorrect way to state a null hypothesis. That’s because when we conduct our inferential tests we’re seeing what the probability is of getting our results even if our null were true. So if we get a p value of say 0.10 (10%), according to the above null hypothesis we’re saying there is a 10% chance that there will be no significant difference between the two conditions, which isn’t actually accurate (don’t worry if I’ve lost you – it’s mind bending stuff). This is one of those instances where poor statistical practice has ingrained itself in IB assessment. But on the plus side it does make it easier for students (and not enough time is spent on this for the bad habits to be too ingrained anyway).

Travis Dixon is an IB Psychology teacher, author, workshop leader, examiner and IA moderator.

Hypothesis testing

• PMID: 8900794
• DOI: 10.1097/00002800-199607000-00009

Hypothesis testing is the process of making a choice between two conflicting hypotheses. The null hypothesis, H0, is a statistical proposition stating that there is no significant difference between a hypothesized value of a population parameter and its value estimated from a sample drawn from that population. The alternative hypothesis, H1 or Ha, is a statistical proposition stating that there is a significant difference between a hypothesized value of a population parameter and its estimated value. When the null hypothesis is tested, a decision is either correct or incorrect. An incorrect decision can be made in two ways: We can reject the null hypothesis when it is true (Type I error) or we can fail to reject the null hypothesis when it is false (Type II error). The probability of making Type I and Type II errors is designated by alpha and beta, respectively. The smallest observed significance level for which the null hypothesis would be rejected is referred to as the p-value. The p-value only has meaning as a measure of confidence when the decision is to reject the null hypothesis. It has no meaning when the decision is that the null hypothesis is true.

• Data Interpretation, Statistical*
• Nursing Research
• Reproducibility of Results

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