hypothesis in data science

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Hypothesis Testing – A Deep Dive into Hypothesis Testing, The Backbone of Statistical Inference

• September 21, 2023

Explore the intricacies of hypothesis testing, a cornerstone of statistical analysis. Dive into methods, interpretations, and applications for making data-driven decisions.

In this Blog post we will learn:

• What is Hypothesis Testing?
• Steps in Hypothesis Testing 2.1. Set up Hypotheses: Null and Alternative 2.2. Choose a Significance Level (α) 2.3. Calculate a test statistic and P-Value 2.4. Make a Decision
• Example : Testing a new drug.
• Example in python

1. What is Hypothesis Testing?

In simple terms, hypothesis testing is a method used to make decisions or inferences about population parameters based on sample data. Imagine being handed a dice and asked if it’s biased. By rolling it a few times and analyzing the outcomes, you’d be engaging in the essence of hypothesis testing.

Think of hypothesis testing as the scientific method of the statistics world. Suppose you hear claims like “This new drug works wonders!” or “Our new website design boosts sales.” How do you know if these statements hold water? Enter hypothesis testing.

2. Steps in Hypothesis Testing

• Set up Hypotheses : Begin with a null hypothesis (H0) and an alternative hypothesis (Ha).
• Choose a Significance Level (α) : Typically 0.05, this is the probability of rejecting the null hypothesis when it’s actually true. Think of it as the chance of accusing an innocent person.
• Calculate Test statistic and P-Value : Gather evidence (data) and calculate a test statistic.
• p-value : This is the probability of observing the data, given that the null hypothesis is true. A small p-value (typically ≤ 0.05) suggests the data is inconsistent with the null hypothesis.
• Decision Rule : If the p-value is less than or equal to α, you reject the null hypothesis in favor of the alternative.

2.1. Set up Hypotheses: Null and Alternative

Before diving into testing, we must formulate hypotheses. The null hypothesis (H0) represents the default assumption, while the alternative hypothesis (H1) challenges it.

For instance, in drug testing, H0 : “The new drug is no better than the existing one,” H1 : “The new drug is superior .”

2.2. Choose a Significance Level (α)

When You collect and analyze data to test H0 and H1 hypotheses. Based on your analysis, you decide whether to reject the null hypothesis in favor of the alternative, or fail to reject / Accept the null hypothesis.

The significance level, often denoted by $α$, represents the probability of rejecting the null hypothesis when it is actually true.

In other words, it’s the risk you’re willing to take of making a Type I error (false positive).

Type I Error (False Positive) :

• Symbolized by the Greek letter alpha (α).
• Occurs when you incorrectly reject a true null hypothesis . In other words, you conclude that there is an effect or difference when, in reality, there isn’t.
• The probability of making a Type I error is denoted by the significance level of a test. Commonly, tests are conducted at the 0.05 significance level , which means there’s a 5% chance of making a Type I error .
• Commonly used significance levels are 0.01, 0.05, and 0.10, but the choice depends on the context of the study and the level of risk one is willing to accept.

Example : If a drug is not effective (truth), but a clinical trial incorrectly concludes that it is effective (based on the sample data), then a Type I error has occurred.

Type II Error (False Negative) :

• Symbolized by the Greek letter beta (β).
• Occurs when you accept a false null hypothesis . This means you conclude there is no effect or difference when, in reality, there is.
• The probability of making a Type II error is denoted by β. The power of a test (1 – β) represents the probability of correctly rejecting a false null hypothesis.

Example : If a drug is effective (truth), but a clinical trial incorrectly concludes that it is not effective (based on the sample data), then a Type II error has occurred.

Balancing the Errors :

In practice, there’s a trade-off between Type I and Type II errors. Reducing the risk of one typically increases the risk of the other. For example, if you want to decrease the probability of a Type I error (by setting a lower significance level), you might increase the probability of a Type II error unless you compensate by collecting more data or making other adjustments.

It’s essential to understand the consequences of both types of errors in any given context. In some situations, a Type I error might be more severe, while in others, a Type II error might be of greater concern. This understanding guides researchers in designing their experiments and choosing appropriate significance levels.

2.3. Calculate a test statistic and P-Value

Test statistic : A test statistic is a single number that helps us understand how far our sample data is from what we’d expect under a null hypothesis (a basic assumption we’re trying to test against). Generally, the larger the test statistic, the more evidence we have against our null hypothesis. It helps us decide whether the differences we observe in our data are due to random chance or if there’s an actual effect.

P-value : The P-value tells us how likely we would get our observed results (or something more extreme) if the null hypothesis were true. It’s a value between 0 and 1. – A smaller P-value (typically below 0.05) means that the observation is rare under the null hypothesis, so we might reject the null hypothesis. – A larger P-value suggests that what we observed could easily happen by random chance, so we might not reject the null hypothesis.

2.4. Make a Decision

Relationship between $α$ and P-Value

When conducting a hypothesis test:

We then calculate the p-value from our sample data and the test statistic.

Finally, we compare the p-value to our chosen $α$:

• If $p−value≤α$: We reject the null hypothesis in favor of the alternative hypothesis. The result is said to be statistically significant.
• If $p−value>α$: We fail to reject the null hypothesis. There isn’t enough statistical evidence to support the alternative hypothesis.

3. Example : Testing a new drug.

Imagine we are investigating whether a new drug is effective at treating headaches faster than drug B.

Setting Up the Experiment : You gather 100 people who suffer from headaches. Half of them (50 people) are given the new drug (let’s call this the ‘Drug Group’), and the other half are given a sugar pill, which doesn’t contain any medication.

• Set up Hypotheses : Before starting, you make a prediction:
• Null Hypothesis (H0): The new drug has no effect. Any difference in healing time between the two groups is just due to random chance.
• Alternative Hypothesis (H1): The new drug does have an effect. The difference in healing time between the two groups is significant and not just by chance.

Calculate Test statistic and P-Value : After the experiment, you analyze the data. The “test statistic” is a number that helps you understand the difference between the two groups in terms of standard units.

For instance, let’s say:

• The average healing time in the Drug Group is 2 hours.
• The average healing time in the Placebo Group is 3 hours.

The test statistic helps you understand how significant this 1-hour difference is. If the groups are large and the spread of healing times in each group is small, then this difference might be significant. But if there’s a huge variation in healing times, the 1-hour difference might not be so special.

Imagine the P-value as answering this question: “If the new drug had NO real effect, what’s the probability that I’d see a difference as extreme (or more extreme) as the one I found, just by random chance?”

For instance:

• P-value of 0.01 means there’s a 1% chance that the observed difference (or a more extreme difference) would occur if the drug had no effect. That’s pretty rare, so we might consider the drug effective.
• P-value of 0.5 means there’s a 50% chance you’d see this difference just by chance. That’s pretty high, so we might not be convinced the drug is doing much.
• If the P-value is less than ($α$) 0.05: the results are “statistically significant,” and they might reject the null hypothesis , believing the new drug has an effect.
• If the P-value is greater than ($α$) 0.05: the results are not statistically significant, and they don’t reject the null hypothesis , remaining unsure if the drug has a genuine effect.

4. Example in python

For simplicity, let’s say we’re using a t-test (common for comparing means). Let’s dive into Python:

Making a Decision : “The results are statistically significant! p-value < 0.05 , The drug seems to have an effect!” If not, we’d say, “Looks like the drug isn’t as miraculous as we thought.”

5. Conclusion

Hypothesis testing is an indispensable tool in data science, allowing us to make data-driven decisions with confidence. By understanding its principles, conducting tests properly, and considering real-world applications, you can harness the power of hypothesis testing to unlock valuable insights from your data.

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9   Hypothesis testing

In scientific studies, you’ll often see phrases like “the results are statistically significant”. This points to a technique called hypothesis testing, where we use p-values, a type of probability, to test our initial assumption or hypothesis.

In hypothesis testing, rather than providing an estimate of the parameter we’re studying, we provide a probability that serves as evidence supporting or contradicting a specific hypothesis. The hypothesis usually involves whether a parameter is different from a predetermined value (often 0).

Hypothesis testing is used when you can phrase your research question in terms of whether a parameter differs from this predetermined value. It’s applied in various fields, asking questions such as: Does a medication extend the lives of cancer patients? Does an increase in gun sales correlate with more gun violence? Does class size affect test scores?

Take, for instance, the previously used example with colored beads. We might not be concerned about the exact proportion of blue beads, but instead ask: Are there more blue beads than red ones? This could be rephrased as asking if the proportion of blue beads is more than 0.5.

The initial hypothesis that the parameter equals the predetermined value is called the “null hypothesis”. It’s popular because it allows us to focus on the data’s properties under this null scenario. Once data is collected, we estimate the parameter and calculate the p-value, which is the probability of the estimate being as extreme as observed if the null hypothesis is true. If the p-value is small, it indicates the null hypothesis is unlikely, providing evidence against it.

We will see more examples of hypothesis testing in Chapter 17 .

9.1 p-values

Suppose we take a random sample of \(N=100\) and we observe \(52\) blue beads, which gives us \(\bar{X} = 0.52\) . This seems to be pointing to the existence of more blue than red beads since 0.52 is larger than 0.5. However, we know there is chance involved in this process and we could get a 52 even when the actual \(p=0.5\) . We call the assumption that \(p = 0.5\) a null hypothesis . The null hypothesis is the skeptic’s hypothesis.

We have observed a random variable \(\bar{X} = 0.52\) , and the p-value is the answer to the question: How likely is it to see a value this large, when the null hypothesis is true? If the p-value is small enough, we reject the null hypothesis and say that the results are statistically significant .

The p-value of 0.05 as a threshold for statistical significance is conventionally used in many areas of research. A cutoff of 0.01 is also used to define highly significance . The choice of 0.05 is somewhat arbitrary and was popularized by the British statistician Ronald Fisher in the 1920s. We do not recommend using these cutoff without justification and recommend avoiding the phrase statistically significant .

To obtain a p-value for our example, we write:

\[\mbox{Pr}(\mid \bar{X} - 0.5 \mid > 0.02 ) \]

assuming the \(p=0.5\) . Under the null hypothesis we know that:

\[ \sqrt{N}\frac{\bar{X} - 0.5}{\sqrt{0.5(1-0.5)}} \]

is standard normal. We, therefore, can compute the probability above, which is the p-value.

\[\mbox{Pr}\left(\sqrt{N}\frac{\mid \bar{X} - 0.5\mid}{\sqrt{0.5(1-0.5)}} > \sqrt{N} \frac{0.02}{ \sqrt{0.5(1-0.5)}}\right)\]

In this case, there is actually a large chance of seeing 52 or larger under the null hypothesis.

Keep in mind that there is a close connection between p-values and confidence intervals. If a 95% confidence interval of the spread does not include 0, we know that the p-value must be smaller than 0.05.

To learn more about p-values, you can consult any statistics textbook. However, in general, we prefer reporting confidence intervals over p-values because it gives us an idea of the size of the estimate. If we just report the p-value, we provide no information about the significance of the finding in the context of the problem.

We can show mathematically that if a \((1-\alpha)\times 100\) % confidence interval does not contain the null hypothesis value, the null hypothesis is rejected with a p-value as smaller or smaller than \(\alpha\) . So statistical significance can be determined from confidence intervals. However, unlike the confidence interval, the p-value does not provide an estimate of the magnitude of the effect. For this reason, we recommend avoiding p-values whenever you can compute a confidence interval.

Pollsters are not successful at providing correct confidence intervals, but rather at predicting who will win. When we took a 25 bead sample size, the confidence interval for the spread:

included 0. If this were a poll and we were forced to make a declaration, we would have to say it was a “toss-up”.

One problem with our poll results is that, given the sample size and the value of \(p\) , we would have to sacrifice the probability of an incorrect call to create an interval that does not include 0.

This does not mean that the election is close. It only means that we have a small sample size. In statistical textbooks, this is called lack of power . In the context of polls, power is the probability of detecting spreads different from 0.

By increasing our sample size, we lower our standard error, and thus, have a much better chance of detecting the direction of the spread.

9.3 Exercises

• Generate a sample of size \(N=1000\) from an urn model with 50% blue beads:

then, compute a p-value to test if \(p=0.5\) . Repeat this 10,000 times and report how often the p-value is lower than 0.05? How often is it lower than 0.01?

• Make a histogram of the p-values you generated in exercise 1. Which of the following seems to be true?
• The p-values are all 0.05.
• The p-values are normally distributed; CLT seems to hold.
• The p-values are uniformly distributed.
• The p-values are all less than 0.05.

Demonstrate, mathematically, why see the histogram we see in exercise 2.

Generate a sample of size \(N=1000\) from an urn model with 52% blue beads:

Compute a p-value to test if \(p=0.5\) . Repeat this 10,000 times and report how often the p-value is larger than 0.05? Note that you are computing 1 - power.

• Repeat exercise for but for the following values:

Plot power as a function of \(N\) with a different color curve for each value of p .

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Statistical Inference and Hypothesis Testing in Data Science Applications

This course is part of Data Science Foundations: Statistical Inference Specialization

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Define a composite hypothesis and the level of significance for a test with a composite null hypothesis., define a test statistic, level of significance, and the rejection region for a hypothesis test. give the form of a rejection region., perform tests concerning a true population variance., compute the sampling distributions for the sample mean and sample minimum of the exponential distribution., details to know.

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There are 6 modules in this course

This course will focus on theory and implementation of hypothesis testing, especially as it relates to applications in data science. Students will learn to use hypothesis tests to make informed decisions from data. Special attention will be given to the general logic of hypothesis testing, error and error rates, power, simulation, and the correct computation and interpretation of p-values. Attention will also be given to the misuse of testing concepts, especially p-values, and the ethical implications of such misuse.

This course can be taken for academic credit as part of CU Boulder’s Master of Science in Data Science (MS-DS) degree offered on the Coursera platform. The MS-DS is an interdisciplinary degree that brings together faculty from CU Boulder’s departments of Applied Mathematics, Computer Science, Information Science, and others. With performance-based admissions and no application process, the MS-DS is ideal for individuals with a broad range of undergraduate education and/or professional experience in computer science, information science, mathematics, and statistics. Learn more about the MS-DS program at https://www.coursera.org/degrees/master-of-science-data-science-boulder.

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Welcome to the course! This module contains logistical information to get you started!

What's included

6 readings 1 app item 1 discussion prompt 1 ungraded lab

6 readings • Total 57 minutes

• Introducing the Yellowdig Learning Community • 10 minutes
• Earn Academic Credit for your Work! • 10 minutes
• Course Support • 10 minutes
• Course Resources • 10 minutes
• Getting Started with Yellowdig • 15 minutes
• Join the Conversation in our Yellowdig Community • 2 minutes

1 app item • Total 60 minutes

• Statistical Inference and Hypothesis Testing in Data Science Applications Yellowdig Community • 60 minutes

1 discussion prompt • Total 10 minutes

• Introduce Yourself • 10 minutes

1 ungraded lab • Total 60 minutes

• Introduction to Jupyter Notebooks and R • 60 minutes

Fundamental Concepts of Hypothesis Testing

In this module, we will define a hypothesis test and develop the intuition behind designing a test. We will learn the language of hypothesis testing, which includes definitions of a null hypothesis, an alternative hypothesis, and the level of significance of a test. We will walk through a very simple test.

6 videos 12 readings 1 quiz 1 programming assignment 2 ungraded labs

6 videos • Total 69 minutes

• What is Hypothesis Testing? • 3 minutes • Preview module
• Types of Hypotheses • 14 minutes
• Normal Computations • 23 minutes
• Errors in Hypothesis Testing • 7 minutes
• Test Statistics and Significance • 14 minutes
• A First Test • 4 minutes

12 readings • Total 107 minutes

• What is Hypothesis Testing? • 5 minutes
• Types of Hypotheses • 10 minutes
• Video Slides for Types of Hypotheses • 10 minutes
• Normal Computations • 10 minutes
• Video Slides for Normal Computations • 10 minutes
• Errors in Hypothesis Testing • 10 minutes
• Video Slides for Errors in Hypothesis Testing • 10 minutes
• Test Statistics and Significance • 10 minutes
• Video Slides for Test Statistics and Level of Significance • 10 minutes
• A First Test • 10 minutes
• Video Slides for A First Test • 10 minutes

1 quiz • Total 30 minutes

• Introduction to Hypothesis Testing • 30 minutes

1 programming assignment • Total 180 minutes

• Intro to Hypothesis Testing Lab • 180 minutes

2 ungraded labs • Total 120 minutes

• An Introduction to R and Jupyter Notebooks • 60 minutes
• Visualizing Errors in Hypothesis Testing • 60 minutes

Composite Tests, Power Functions, and P-Values

In this module, we will expand the lessons of Module 1 to composite hypotheses for both one and two-tailed tests. We will define the “power function” for a test and discuss its interpretation and how it can lead to the idea of a “uniformly most powerful” test. We will discuss and interpret “p-values” as an alternate approach to hypothesis testing.

7 videos 8 readings 1 quiz 1 programming assignment 1 ungraded lab

7 videos • Total 124 minutes

• Composite Hypotheses and Level of Significance • 16 minutes • Preview module
• One-Tailed Tests • 20 minutes
• Power Functions • 13 minutes
• Hypothesis Testing with P-Values • 21 minutes
• Two Tailed Tests • 12 minutes
• CLT: A Brief Review • 16 minutes
• Hypothesis Tests for Proportions • 23 minutes

8 readings • Total 72 minutes

• Video Slides for Composite Hypotheses and Level of Significance • 10 minutes
• Video Slides for One-Tailed Tests • 10 minutes
• Video Slides for Power Functions • 10 minutes
• Video Slides for Hypothesis Testing with P-Values • 10 minutes
• Video Slides for Two-Tailed Tests • 10 minutes
• Video Slides for CLT: A Brief Review • 10 minutes
• Video Slides for Hypothesis Tests for Proportions • 10 minutes
• Constructing Tests • 30 minutes
• The Basics of Hypothesis Testing • 180 minutes
• Distributions of P-Values • 60 minutes

t-Tests and Two-Sample Tests

In this module, we will learn about the chi-squared and t distributions and their relationships to sampling distributions. We will learn to identify when hypothesis tests based on these distributions are appropriate. We will review the concept of sample variance and derive the “t-test”. Additionally, we will derive our first two-sample test and apply it to make some decisions about real data.

7 videos • Total 139 minutes

• The t and Chi-Squared Distributions • 41 minutes • Preview module
• The Sample Variance for the Normal Distribution • 23 minutes
• t-Tests • 18 minutes
• Two Sample Tests for Means • 15 minutes
• Two Sample t-Tests for a Difference of Means • 17 minutes
• Welch's t-Test and Paired Data • 13 minutes
• Comparing Population Proportions • 8 minutes
• Video Slides for the t and Chi-Squared Distributions • 10 minutes
• Video Slides for the Sample Variance and the Normal Distribution • 10 minutes
• Video Slides for t-Tests • 10 minutes
• Video Slides for Two Sample Tests for Means • 10 minutes
• Video Slides for Differences in Population Means • 10 minutes
• Video Slides for Welch's Test and Paired Data • 10 minutes
• Video Slides for Comparing Population Proportions • 10 minutes
• More Hypothesis Tests! • 30 minutes
• t-Tests • 180 minutes
• t-Tests and Two Sample Tests • 60 minutes

Beyond Normality

In this module, we will consider some problems where the assumption of an underlying normal distribution is not appropriate and will expand our ability to construct hypothesis tests for this case. We will define the concept of a “uniformly most powerful” (UMP) test, whether or not such a test exists for specific problems, and we will revisit some of our earlier tests from Modules 1 and 2 through the UMP lens. We will also introduce the F-distribution and its role in testing whether or not two population variances are equal.

6 videos 7 readings 2 quizzes

6 videos • Total 117 minutes

• Properties of the Exponential Distribution • 13 minutes • Preview module
• Two Tests • 27 minutes
• Best Tests • 22 minutes
• UMP Tests • 10 minutes
• A Test for the Variance of the Normal Distribution • 12 minutes
• The F-Distribution and a Ratio of Variances • 31 minutes

7 readings • Total 62 minutes

• Video Slides for Properties of the Exponential Distribution • 10 minutes
• Video Slides for Two Hypothesis Tests for the Exponential • 10 minutes
• Video Slides for Best Tests • 10 minutes
• Video Slides for UMP Tests • 10 minutes
• Video Slides for a Normal Variance Test • 10 minutes
• Video Slides for an F-Distribution and a Ratio of Variances • 10 minutes

2 quizzes • Total 60 minutes

• Best Tests and Some General Skills • 30 minutes
• Uniformly Most Powerful Tests and F-Tests • 30 minutes

Likelihood Ratio Tests and Chi-Squared Tests

In this module, we develop a formal approach to hypothesis testing, based on a “likelihood ratio” that can be more generally applied than any of the tests we have discussed so far. We will pay special attention to the large sample properties of the likelihood ratio, especially Wilks’ Theorem, that will allow us to come up with approximate (but easy) tests when we have a large sample size. We will close the course with two chi-squared tests that can be used to test whether the distributional assumptions we have been making throughout this course are valid.

5 videos 7 readings 1 quiz 1 programming assignment 1 ungraded lab

5 videos • Total 93 minutes

• MLEs • 23 minutes • Preview module
• The GRLT • 15 minutes
• Wilks' Theorem • 12 minutes
• Chi-Squared Goodness of Fit Test • 23 minutes
• Independence and Homogeneity • 19 minutes
• Video Slides for MLEs • 10 minutes
• Video Slides for the GLRT • 10 minutes
• Video Slides for Wilks' Theorem • 10 minutes
• Video Slides for Chi-Squared Goodness of Fit Test • 10 minutes
• Video Slides for Independence and Homogeneity • 10 minutes
• Share Your Feedback on Yellowdig • 10 minutes
• Adventures in GLRTs • 30 minutes
• Chi-Squared Tests and Mo • 180 minutes
• Exploring Wilks' Theorem • 60 minutes

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Hypothesis Testing in Data Science: It's Usage and Types

Hypothesis Testing in Data Science is a crucial method for making informed decisions from data. This blog explores its essential usage in analysing trends and patterns, and the different types such as null, alternative, one-tailed, and two-tailed tests, providing a comprehensive understanding for both beginners and advanced practitioners.

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Table of Contents  

1) What is Hypothesis Testing in Data Science? 

2) Importance of Hypothesis Testing in Data Science 

3) Types of Hypothesis Testing 

4) Basic steps in Hypothesis Testing 

5) Real-world use cases of Hypothesis Testing 

6) Conclusion 

What is Hypothesis Testing in Data Science?  

Hypothesis Testing in Data Science is a statistical method used to assess the validity of assumptions or claims about a population based on sample data. It involves formulating two Hypotheses, the null Hypothesis (H0) and the alternative Hypothesis (Ha or H1), and then using statistical tests to find out if there is enough evidence to support the alternative Hypothesis.  

Hypothetical Testing is a critical tool for making data-driven decisions, evaluating the significance of observed effects or differences, and drawing meaningful conclusions from data, allowing Data Scientists to uncover patterns, relationships, and insights that inform various domains, from medicine to business and beyond. 

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Importance of Hypothesis Testing in Data Science  

The significance of Hypothesis Testing in Data Science cannot be overstated. It serves as the cornerstone of data-driven decision-making. By systematically testing Hypotheses, Data Scientists can: 

Objective decision-making 

Hypothesis Testing provides a structured and impartial method for making decisions based on data. In a world where biases can skew perceptions, Data Scientists rely on this method to ensure that their conclusions are grounded in empirical evidence, making their decisions more objective and trustworthy. 

Statistical rigour 

Data Scientists deal with large amounts of data, and Hypothesis Testing helps them make sense of it. It quantifies the significance of observed patterns, differences, or relationships. This statistical rigour is essential in distinguishing between mere coincidences and meaningful findings, reducing the likelihood of making decisions based on random chance. 

Resource allocation 

Resources, whether they are financial, human, or time-related, are often limited. Hypothesis Testing enables efficient resource allocation by guiding Data Scientists towards strategies or interventions that are statistically significant. This ensures that efforts are directed where they are most likely to yield valuable results. 

Risk management 

In domains like healthcare and finance, where lives and livelihoods are at stake, Hypothesis Testing is a critical tool for risk assessment. For instance, in drug development, Hypothesis Testing is used to determine the safety and efficiency of new treatments, helping mitigate potential risks to patients. 

Innovation and progress 

Hypothesis Testing fosters innovation by providing a systematic framework to evaluate new ideas, products, or strategies. It encourages a cycle of experimentation, feedback, and improvement, driving continuous progress and innovation. 

Strategic decision-making 

Organisations base their strategies on data-driven insights. Hypothesis Testing enables them to make informed decisions about market trends, customer behaviour, and product development. These decisions are grounded in empirical evidence, increasing the likelihood of success. 

Scientific integrity 

In scientific research, Hypothesis Testing is integral to maintaining the integrity of research findings. It ensures that conclusions are drawn from rigorous statistical analysis rather than conjecture. This is essential for advancing knowledge and building upon existing research. 

Regulatory compliance 

Many industries, such as pharmaceuticals and aviation, operate under strict regulatory frameworks. Hypothesis Testing is essential for demonstrating compliance with safety and quality standards. It provides the statistical evidence required to meet regulatory requirements. 

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Types of Hypothesis Testing  

Hypothesis Testing can be seen in several different types. In total, we have five types of Hypothesis Testing. They are described below as follows: 

Alternative Hypothesis

The Alternative Hypothesis, denoted as Ha or H1, is the assertion or claim that researchers aim to support with their data analysis. It represents the opposite of the null Hypothesis (H0) and suggests that there is a significant effect, relationship, or difference in the population. In simpler terms, it's the statement that researchers hope to find evidence for during their analysis. For example, if you are testing a new drug's efficacy, the alternative Hypothesis might state that the drug has a measurable positive effect on patients' health. 

Null Hypothesis 

The Null Hypothesis, denoted as H0, is the default assumption in Hypothesis Testing. It posits that there is no significant effect, relationship, or difference in the population being studied. In other words, it represents the status quo or the absence of an effect. Researchers typically set out to challenge or disprove the Null Hypothesis by collecting and analysing data. Using the drug efficacy example again, the Null Hypothesis might state that the new drug has no effect on patients' health. 

Non-directional Hypothesis 

A Non-directional Hypothesis, also known as a two-tailed Hypothesis, is used when researchers are interested in whether there is any significant difference, effect, or relationship in either direction (positive or negative). This type of Hypothesis allows for the possibility of finding effects in both directions. For instance, in a study comparing the performance of two groups, a Non-directional Hypothesis would suggest that there is a significant difference between the groups, without specifying which group performs better. 

Directional Hypothesis 

A Directional Hypothesis, also called a one-tailed Hypothesis, is employed when researchers have a specific expectation about the direction of the effect, relationship, or difference they are investigating. In this case, the Hypothesis predicts an outcome in a particular direction—either positive or negative. For example, if you expect that a new teaching method will improve student test scores, a directional Hypothesis would state that the new method leads to higher test scores. 

Statistical Hypothesis 

A Statistical Hypothesis is a Hypothesis formulated in a way that it can be tested using statistical methods. It involves specific numerical values or parameters that can be measured or compared. Statistical Hypotheses are crucial for quantitative research and often involve means, proportions, variances, correlations, or other measurable quantities. These Hypotheses provide a precise framework for conducting statistical tests and drawing conclusions based on data analysis. 

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Basic steps in Hypothesis Testing  

Hypothesis Testing is a systematic approach used in statistics to make informed decisions based on data. It is a critical tool in Data Science, research, and many other fields where data analysis is employed. The following are the basic steps involved in Hypothesis Testing: 

1) Formulate Hypotheses 

The first step in Hypothesis Testing is to clearly define your research question and translate it into two mutually exclusive Hypotheses: 

a) Null Hypothesis (H0): This is the default assumption, often representing the status quo or the absence of an effect. It states that there is no significant difference, relationship, or effect in the population. 

b) Alternative Hypothesis (Ha or H1): This is the statement that contradicts the null Hypothesis. It suggests that there is a significant difference, relationship, or effect in the population. 

The formulation of these Hypotheses is crucial, as they serve as the foundation for your entire Hypothesis Testing process. 

2) Collect data 

With your Hypotheses in place, the next step is to gather relevant data through surveys, experiments, observations, or any other suitable method. The data collected should be representative of the population you are studying. The quality and quantity of data are essential factors in the success of your Hypothesis Testing. 

3) Choose a significance level (α) 

Before conducting the statistical test, you need to decide on the level of significance, denoted as α. The significance level represents the threshold for statistical significance and determines how confident you want to be in your results. A common choice is α = 0.05, which implies a 5% chance of making a Type I error (rejecting the null Hypothesis when it's true). You can choose a different α value based on the specific requirements of your analysis. 

4) Perform the test 

Based on the nature of your data and the Hypotheses you've formulated, select the appropriate statistical test. There are various tests available, including t-tests, chi-squared tests, ANOVA, regression analysis, and more. The chosen test should align with the type of data (e.g., continuous or categorical) and the research question (e.g., comparing means or testing for independence). 

Execute the selected statistical test on your data to obtain test statistics and p-values. The test statistics quantify the difference or effect you are investigating, while the p-value represents the probability of obtaining the observed results if the null Hypothesis were true. 

5) Analyse the results 

Once you have the test statistics and p-value, it's time to interpret the results. The primary focus is on the p-value: 

a) If the p-value is less than or equal to your chosen significance level (α), typically 0.05, you have evidence to reject the null Hypothesis. This shows that there is a significant difference, relationship, or effect in the population. 

b) If the p-value is more than α, you fail to reject the null Hypothesis, showing that there is insufficient evidence to support the alternative Hypothesis. 

6) Draw conclusions 

Based on the analysis of the p-value and the comparison to the significance level, you can draw conclusions about your research question: 

a) In case you reject the null Hypothesis, you can accept the alternative Hypothesis and make inferences based on the evidence provided by your data. 

b) In case you fail to reject the null Hypothesis, you do not accept the alternative Hypothesis, and you acknowledge that there is no significant evidence to support your claim. 

It's important to communicate your findings clearly, including the implications and limitations of your analysis. 

Real-world use cases of Hypothesis Testing  

The following are some of the real-world use cases of Hypothesis Testing. 

a) Medical research: Hypothesis Testing is crucial in determining the efficacy of new medications or treatments. For instance, in a clinical trial, researchers use Hypothesis Testing to assess whether a new drug is significantly more effective than a placebo in treating a particular condition. 

b) Marketing and advertising: Businesses employ Hypothesis Testing to evaluate the impact of marketing campaigns. A company may test whether a new advertising strategy leads to a significant increase in sales compared to the previous approach. 

c) Manufacturing and quality control: Manufacturing industries use Hypothesis Testing to ensure product quality. For example, in the automotive industry, Hypothesis Testing can be applied to test whether a new manufacturing process results in a significant reduction in defects. 

d) Education: In the field of education, Hypothesis Testing can be used to assess the effectiveness of teaching methods. Researchers may test whether a new teaching approach leads to statistically significant improvements in student performance. 

e) Finance and investment: Investment strategies are often evaluated using Hypothesis Testing. Investors may test whether a new investment strategy outperforms a benchmark index over a specified period.  

Conclusion 

To sum it up, Hypothesis Testing in Data Science is a powerful tool that enables Data Scientists to make evidence-based decisions and draw meaningful conclusions from data. Understanding the types, methods, and steps involved in Hypothesis Testing is essential for any Data Scientist. By rigorously applying Hypothesis Testing techniques, you can gain valuable insights and drive informed decision-making in various domains. 

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Hypothesis Testing in Data Science [Types, Process, Example]

Home Blog Data Science Hypothesis Testing in Data Science [Types, Process, Example]

In day-to-day life, we come across a lot of data lot of variety of content. Sometimes the information is too much that we get confused about whether the information provided is correct or not. At that moment, we get introduced to a word called “Hypothesis testing” which helps in determining the proofs and pieces of evidence for some belief or information.  

What is Hypothesis Testing?

Hypothesis testing is an integral part of statistical inference. It is used to decide whether the given sample data from the population parameter satisfies the given hypothetical condition. So, it will predict and decide using several factors whether the predictions satisfy the conditions or not. In simpler terms, trying to prove whether the facts or statements are true or not.   

For example, if you predict that students who sit on the last bench are poorer and weaker than students sitting on 1st bench, then this is a hypothetical statement that needs to be clarified using different experiments. Another example we can see is implementing new business strategies to evaluate whether they will work for the business or not. All these things are very necessary when you work with data as a data scientist.  If you are interested in learning about data science, visit this amazing  Data Science full course   to learn data science.    

How is Hypothesis Testing Used in Data Science?

It is important to know how and where we can use hypothesis testing techniques in the field of data science. Data scientists predict a lot of things in their day-to-day work, and to check the probability of whether that finding is certain or not, we use hypothesis testing. The main goal of hypothesis testing is to gauge how well the predictions perform based on the sample data provided by the population. If you are interested to know more about the applications of the data, then refer to this  D ata  Scien ce course in India  which will give you more insights into application-based things. When data scientists work on model building using various machine learning algorithms, they need to have faith in their models and the forecasting of models. They then provide the sample data to the model for training purposes so that it can provide us with the significance of statistical data that will represent the entire population.  

Where and When to Use Hypothesis Test?

Hypothesis testing is widely used when we need to compare our results based on predictions. So, it will compare before and after results. For example, someone claimed that students writing exams from blue pen always get above 90%; now this statement proves it correct, and experiments need to be done. So, the data will be collected based on the student's input, and then the test will be done on the final result later after various experiments and observations on students' marks vs pen used, final conclusions will be made which will determine the results. Now hypothesis testing will be done to compare the 1st and the 2nd result, to see the difference and closeness of both outputs. This is how hypothesis testing is done.  

How Does Hypothesis Testing Work in Data Science?

In the whole data science life cycle, hypothesis testing is done in various stages, starting from the initial part, the 1st stage where the EDA, data pre-processing, and manipulation are done. In this stage, we will do our initial hypothesis testing to visualize the outcome in later stages. The next test will be done after we have built our model, once the model is ready and hypothesis testing is done, we will compare the results of the initial testing and the 2nd one to compare the results and significance of the results and to confirm the insights generated from the 1st cycle match with the 2nd one or not. This will help us know how the model responds to the sample training data. As we saw above, hypothesis testing is always needed when we are planning to contrast more than 2 groups. While checking on the results, it is important to check on the flexibility of the results for the sample and the population. Later, we can judge on the disagreement of the results are appropriate or vague. This is all we can do using hypothesis testing.   

Different Types of Hypothesis Testing

Hypothesis testing can be seen in several types. In total, we have 5 types of hypothesis testing. They are described below:

1. Alternative Hypothesis

The alternative hypothesis explains and defines the relationship between two variables. It simply indicates a positive relationship between two variables which means they do have a statistical bond. It indicates that the sample observed is going to influence or affect the outcome. An alternative hypothesis is described using H a  or H 1 . Ha indicates an alternative hypothesis and H 1  explains the possibility of influenced outcome which is 1. For example, children who study from the beginning of the class have fewer chances to fail. An alternate hypothesis will be accepted once the statistical predictions become significant. The alternative hypothesis can be further divided into 3 parts.   

• Left-tailed: Left tailed hypothesis can be expected when the sample value is less than the true value.   
• Right-tailed: Right-tailed hypothesis can be expected when the true value is greater than the outcome/predicted value.    
• Two-tailed: Two-tailed hypothesis is defined when the true value is not equal to the sample value or the output.   

2. Null Hypothesis

The null hypothesis simply states that there is no relation between statistical variables. If the facts presented at the start do not match with the outcomes, then we can say, the testing is null hypothesis testing. The null hypothesis is represented as H 0 . For example, children who study from the beginning of the class have no fewer chances to fail. There are types of Null Hypothesis described below:   

Simple Hypothesis:  It helps in denoting and indicating the distribution of the population.   

Composite Hypothesis:  It does not denote the population distribution   

Exact Hypothesis:  In the exact hypothesis, the value of the hypothesis is the same as the sample distribution. Example- μ= 10   

Inexact Hypothesis:  Here, the hypothesis values are not equal to the sample. It will denote a particular range of values.   

3. Non-directional Hypothesis 

The non-directional hypothesis is a tow-tailed hypothesis that indicates the true value does not equal the predicted value. In simpler terms, there is no direction between the 2 variables. For an example of a non-directional hypothesis, girls and boys have different methodologies to solve a problem. Here the example explains that the thinking methodologies of a girl and a boy is different, they don’t think alike.    

4. Directional Hypothesis

In the Directional hypothesis, there is a direct relationship between two variables. Here any of the variables influence the other.   

5. Statistical Hypothesis

Statistical hypothesis helps in understanding the nature and character of the population. It is a great method to decide whether the values and the data we have with us satisfy the given hypothesis or not. It helps us in making different probabilistic and certain statements to predict the outcome of the population... We have several types of tests which are the T-test, Z-test, and Anova tests.  

Methods of Hypothesis Testing

1. frequentist hypothesis testing.

Frequentist hypotheses mostly work with the approach of making predictions and assumptions based on the current data which is real-time data. All the facts are based on current data. The most famous kind of frequentist approach is null hypothesis testing.    

2. Bayesian Hypothesis Testing

Bayesian testing is a modern and latest way of hypothesis testing. It is known to be the test that works with past data to predict the future possibilities of the hypothesis. In Bayesian, it refers to the prior distribution or prior probability samples for the observed data. In the medical Industry, we observe that Doctors deal with patients’ diseases using past historical records. So, with this kind of record, it is helpful for them to understand and predict the current and upcoming health conditions of the patient.

Importance of Hypothesis Testing in Data Science

Most of the time, people assume that data science is all about applying machine learning algorithms and getting results, that is true but in addition to the fact that to work in the data science field, one needs to be well versed with statistics as most of the background work in Data science is done through statistics. When we deal with data for pre-processing, manipulating, and analyzing, statistics play. Specifically speaking Hypothesis testing helps in making confident decisions, predicting the correct outcomes, and finding insightful conclusions regarding the population. Hypothesis testing helps us resolve tough things easily. To get more familiar with Hypothesis testing and other prediction models attend the superb useful  KnowledgeHut Data Science full course  which will give you more domain knowledge and will assist you in working with industry-related projects.          

Basic Steps in Hypothesis Testing [Workflow]

1. null and alternative hypothesis.

After we have done our initial research about the predictions that we want to find out if true, it is important to mention whether the hypothesis done is a null hypothesis(H0) or an alternative hypothesis (Ha). Once we understand the type of hypothesis, it will be easy for us to do mathematical research on it. A null hypothesis will usually indicate the no-relationship between the variables whereas an alternative hypothesis describes the relationship between 2 variables.    

• H0 – Girls, on average, are not strong as boys   
• Ha - Girls, on average are stronger than boys   

2. Data Collection

To prove our statistical test validity, it is essential and critical to check the data and proceed with sampling them to get the correct hypothesis results. If the target data is not prepared and ready, it will become difficult to make the predictions or the statistical inference on the population that we are planning to make. It is important to prepare efficient data, so that hypothesis findings can be easy to predict.   

3. Selection of an appropriate test statistic

To perform various analyses on the data, we need to choose a statistical test. There are various types of statistical tests available. Based on the wide spread of the data that is variance within the group or how different the data category is from one another that is variance without a group, we can proceed with our further research study.   

4. Selection of the appropriate significant level

Once we get the result and outcome of the statistical test, we have to then proceed further to decide whether the reject or accept the null hypothesis. The significance level is indicated by alpha (α). It describes the probability of rejecting or accepting the null hypothesis. Example- Suppose the value of the significance level which is alpha is 0.05. Now, this value indicates the difference from the null hypothesis. 

5. Calculation of the test statistics and the p-value

P value is simply the probability value and expected determined outcome which is at least as extreme and close as observed results of a hypothetical test. It helps in evaluating and verifying hypotheses against the sample data. This happens while assuming the null hypothesis is true. The lower the value of P, the higher and better will be the results of the significant value which is alpha (α). For example, if the P-value is 0.05 or even less than this, then it will be considered statistically significant. The main thing is these values are predicted based on the calculations done by deviating the values between the observed one and referenced one. The greater the difference between values, the lower the p-value will be.

6. Findings of the test

After knowing the P-value and statistical significance, we can determine our results and take the appropriate decision of whether to accept or reject the null hypothesis based on the facts and statistics presented to us.

How to Calculate Hypothesis Testing?

Hypothesis testing can be done using various statistical tests. One is Z-test. The formula for Z-test is given below:  

            Z = ( x̅  – μ 0 )  / (σ /√n)    

In the above equation, x̅ is the sample mean   

• μ0 is the population mean   
• σ is the standard deviation    
• n is the sample size   

Now depending on the Z-test result, the examination will be processed further. The result is either going to be a null hypothesis or it is going to be an alternative hypothesis. That can be measured through below formula-   

• H0: μ=μ0   
• Ha: μ≠μ0   
• Here,   
• H0 = null hypothesis   
• Ha = alternate hypothesis   

In this way, we calculate the hypothesis testing and can apply it to real-world scenarios.

Real-World Examples of Hypothesis Testing

Hypothesis testing has a wide variety of use cases that proves to be beneficial for various industries.    

1. Healthcare

In the healthcare industry, all the research and experiments which are done to predict the success of any medicine or drug are done successfully with the help of Hypothesis testing.   

2. Education sector

Hypothesis testing assists in experimenting with different teaching techniques to deal with the understanding capability of different students.   

3. Mental Health

Hypothesis testing helps in indicating the factors that may cause some serious mental health issues.   

4. Manufacturing

Testing whether the new change in the process of manufacturing helped in the improvement of the process as well as in the quantity or not.  In the same way, there are many other use cases that we get to see in different sectors for hypothesis testing. 

Error Terms in Hypothesis Testing

1. type-i error.

Type I error occurs during the process of hypothesis testing when the null hypothesis is rejected even though it is accurate. This kind of error is also known as False positive because even though the statement is positive or correct but results are given as false. For example, an innocent person still goes to jail because he is considered to be guilty.   

2. Type-II error

Type II error occurs during the process of hypothesis testing when the null hypothesis is not rejected even though it is inaccurate. This Kind of error is also called a False-negative which means even though the statements are false and inaccurate, it still says it is correct and doesn’t reject it. For example, a person is guilty, but in court, he has been proven innocent where he is guilty, so this is a Type II error.   

3. Level of Significance

The level of significance is majorly used to measure the confidence with which a null hypothesis can be rejected. It is the value with which one can reject the null hypothesis which is H0. The level of significance gauges whether the hypothesis testing is significant or not.   

P-value stands for probability value, which tells us the probability or likelihood to find the set of observations when the null hypothesis is true using statistical tests. The main purpose is to check the significance of the statistical statement.   

5. High P-Values

A higher P-value indicates that the testing is not statistically significant. For example, a P value greater than 0.05 is considered to be having higher P value. A higher P-value also means that our evidence and proofs are not strong enough to influence the population.

In hypothesis testing, each step is responsible for getting the outcomes and the results, whether it is the selection of statistical tests or working on data, each step contributes towards the better consequences of the hypothesis testing. It is always a recommendable step when planning for predicting the outcomes and trying to experiment with the sample; hypothesis testing is a useful concept to apply.   

Frequently Asked Questions (FAQs)

We can test a hypothesis by selecting a correct hypothetical test and, based on those getting results.   

Many statistical tests are used for hypothetical testing which includes Z-test, T-test, etc. 

Hypothesis helps us in doing various experiments and working on a specific research topic to predict the results.   

The null and alternative hypothesis, data collection, selecting a statistical test, selecting significance value, calculating p-value, check your findings.    

In simple words, parametric tests are purely based on assumptions whereas non-parametric tests are based on data that is collected and acquired from a sample.   

Gauri Guglani

Gauri Guglani works as a Data Analyst at Deloitte Consulting. She has done her major in Information Technology and holds great interest in the field of data science. She owns her technical skills as well as managerial skills and also is great at communicating. Since her undergraduate, Gauri has developed a profound interest in writing content and sharing her knowledge through the manual means of blog/article writing. She loves writing on topics affiliated with Statistics, Python Libraries, Machine Learning, Natural Language processes, and many more.

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Introduction to Data Science I & II

Hypothesis testing, hypothesis testing #.

Dan L. Nicolae

Hypothesis testing can be thought of as a way to investigate the consistency of a dataset with a model, where a model is a set of rules that describe how data are generated. The consistency is evaluated using ideas from probability and probability distributions.

The consistency question in the above diagram is short for “Is it plausible that data was generated from this model?”

We will use a simple example to illustrate this. Suppose that a friend is telling you that she has an urn with 6 blue and 4 red balls from which 5 balls are extracted without replacement. The description in the previous sentence is that of a model with four rules:

there is an urn with 10 balls: 6 blue and 4 red;

a total of 5 balls are extracted;

the balls are extracted without replacement (once a ball is out of the urn, it cannot be selected again);

at each extraction, every ball in the urn has the same chance of being selected (this assumption is implicit in urn problems).

Suppose your friend reports the results of a drawing (these are the data) and here are two hypothetical scenarios (datasets):

Scenario 1: outcome is 5 red balls . Is this outcome consistent with the model above? The answer is clearly no as it is not possible to obtain 5 red balls when the first 3 rules above are true.

Scenario 2: outcome is 2 blue and 3 red balls . The answer here is not as obvious as above, but we can use probability to get an evaluation of how likely this outcome is. We will formalize this process in this chapter.

We will use these ideas in the next sections to answer questions that are more complicated: Is pollution associated with risk of cancer? Are weights of babies different for older mothers?

We end this introduction with examples of other data-generating models (so you can gain more insight before learning how to evaluate them):

A simple random sample of 10 voters from a population of size 10,000 where 40% of the subjects vote for candidate A, 35% for candidate B and 25% for candidate C.

Data from a binomial setting; this was introduced in the previous chapter where the binomial distribution comes from a sequence of Bernoulli trials that follow 4 rules: (i) a fixed number of trials; (ii) two possible outcomes for each trial; (iii) trials are independent; and (iv) the probability of success is the same for each trial

A set of 100 observations generated independently from a Unif(1,5) distribution.

Hypothesis Testing

Hypothesis Tests (or Significance Tests) are statistical tests to see if a difference we observe is due to chance.

There are many different types of hypothesis tests for different scenarios, but they all have the same basic ideas. Below are the general steps to performing a hypothesis test:

• Formulate your Null and Alternative Hypotheses.
• Ho- Null Hypothesis : The null hypothesis is the hypothesis of no effect. It's the dull, boring hypothesis that says that nothing interesting is going on. If we are trying to test if a difference we observe is due to chance, the null says it is!
• Ha- Alternative Hypothesis : The alternative hypothesis is the opposite of the null. It's what you are trying to test. If we are trying to test if a difference we observe is due to chance, the alternative says it is not!

Think about what you would expect to get if you randomly sampled from the population, assuming the null is true. Compare your observed data and expected data and calculate the test statistic .

Calculate the probability of getting the data you got or something even more extreme if the null were true. This is called the p-value .

• Make your conclusion and interpret it in the context of the problem. If p is very low, we say that the data support rejecting the null hypothesis.

How low is “very low”?

The convention is to reject the null when P < 5% (P < 0.05) and call the result “significant”. There’s no particular justification for this value but it’s commonly used.

The P-value cut-off is called the significance level and is often represented by the Greek letter alpha (α).

The One Sample Z Test: One-sided Hypothesis

The first type of hypothesis test we are going to look at is the one-sample z-test. You can do a z-test for means or for proportions. This is the most simple type of hypothesis test and it uses z-scores and the normal curve. Let’s look at one below!

Hypothesis Test Example : Suppose a large university claims that the average ACT score of their incoming freshman class is 30, but we think the University may be inflating their average. To test the University’s claim we take a simple random sample of 50 students and find their average to be only 28.3 with an SD of 4. Perform a hypothesis test to test the claim. Here are the 4 steps:

• This can be written in symbols as well: Ho: μ = 30
• μ is the symbol for the population mean
• This can be written in symbols as well: Ho: μ < 30
• Our test statistic for the one sample z test is z! We can calculate z using our z-score formula for random variables since we are dealing with a sample of 50 students.

• In our case, the expected value (EV) is 30 since we are assuming our null hypothesis is true (until proven otherwise).
• Since we are dealing with means, our SE is found using the following formula:

Our z-score is -3. See the calculation below:

• Calculate the probability of getting the data you got or something even more extreme if the null were true. This is called the p-value . In this case, our p-value is going to be the area to the left of z = -3. We can use Python to calculate this by using norm.cdf(-3).
• We get that the p-value is 0.0013.
• This is the probability that we would get a sample average of 28.3 given that the null hypothesis was true (the true average was 30).

• Our p-value is less than 5% so we reject our Null Hypothesis. In other words, there is evidence of the Alternative Hypothesis (that the University is inflating their average).

The One Sample Z Test: Two-sided Hypothesis

Hypothesis Test Example : Now we're going to test the above claim but with a different alternative hypothesis. The large university still claims that the average ACT score of their incoming freshman class is 30, but now we think the University may be inflating or deflating their average. To test the University’s claim we take a simple random sample of 50 students and find their average to be only 28.3 with an SD of 4. Perform a hypothesis test to test the claim with our new alternative hypothesis. Here are the 4 steps:

• This can be written in symbols as well: Ho: μ != 30

Step 2 is the same as the one-sided example, so our z score is still -3.

Calculate the probability of getting the data you got or something even more extreme if the null were true. This is called the p-value . In this case, our p-value is going to be the area to the left or right of z = -3. We can use Python to calculate this by using 2*norm.cdf(-3).

• We get that the p-value is 0.0027.
• Our p-value is less than 5% so we reject our Null Hypothesis. In other words, there is evidence of the Alternative Hypothesis (that the University is inflating or deflating their average).

Example Walk-Throughs with Worksheets

Video 1: one sample z-test examples.

• Download Blank Worksheet (PDF)

Video 2: Two Sample z-test Examples

Video 3: z-tests in Python

Video 4: One Sample t-test Examples

Video 5: t-tests in Python

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Tutorial: Statistical Tests of Hypothesis

• February 19, 2019 at 4:30 am

This article is a solid introduction to statistical testing, for beginners, as well as a reference for practitioners. It includes numerous examples as well as illustrations and definitions for concepts such as rejecting the null hypothesis, one sample hypothesis testing, P-values, critical values, and Bayesian hypothesis testing. It has references to additional topics, such as 

• What is Ad Hoc Testing?
• What is a Rejection Region?
• What is a Two Tailed Test?
• How to Decide if a Hypothesis Test is a One Tailed Test or a Two Tailed Test.
• How to Decide if a Hypothesis is a Left Tailed Test or a Right-Tailed Test.
• How to State the Null Hypothesis in Statistics.
• How to Find a Critical Value.
• How to Support or Reject a Null Hypothesis.

Picture: When to use ANOVA?

You can read this tutorial here . For specific popular tests, check the following links below, from the same source:

• Chi Square Test for Normality
• Cochran-Mantel-Haenszel Test
• Granger Causality Test .
• Hotelling’s T-Squared
• KPSS Test .
• What is a Likelihood-Ratio Test?
• Log rank test .
• Sequential Probability Ratio Test
• How to Run a Sign Test.
• T Test: one sample.
• T-Test: Two sample .
• Welch’s ANOVA .
• Welch’s Test for Unequal Variances .
• Z-Test: one sample .
• Z Test: Two Proportion
• Wald Test .

For non standard tests, follow this link . 

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Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.

What is Hypothesis Testing?

Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. 

Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.

Defining Hypotheses

Key Terms of Hypothesis Testing

• P-value: The P value , or calculated probability, is the probability of finding the observed/extreme results when the null hypothesis(H0) of a study-given problem is true. If your P-value is less than the chosen significance level then you reject the null hypothesis i.e. accept that your sample claims to support the alternative hypothesis.
• Test Statistic: The test statistic is a numerical value calculated from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It is compared to a critical value or p-value to make decisions about the statistical significance of the observed results.
• Critical value : The critical value in statistics is a threshold or cutoff point used to determine whether to reject the null hypothesis in a hypothesis test.
• Degrees of freedom: Degrees of freedom are associated with the variability or freedom one has in estimating a parameter. The degrees of freedom are related to the sample size and determine the shape.

Why do we use Hypothesis Testing?

Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing. 

One-Tailed and Two-Tailed Test

One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.

One-Tailed Test

There are two types of one-tailed test:

Two-Tailed Test

A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.

What are Type 1 and Type 2 errors in Hypothesis Testing?

In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.

How does Hypothesis Testing work?

Step 1: define null and alternative hypothesis.

We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.

Step 2 – Choose significance level

Step 3 – Collect and Analyze data.

Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.

Step 4-Calculate Test Statistic

The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.

There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.

• Z-test : If population means and standard deviations are known. Z-statistic is commonly used.
• t-test : If population standard deviations are unknown. and sample size is small than t-test statistic is more appropriate.
• Chi-square test : Chi-square test is used for categorical data or for testing independence in contingency tables
• F-test : F-test is often used in analysis of variance (ANOVA) to compare variances or test the equality of means across multiple groups.

We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.

T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

Step 5 – Comparing Test Statistic:

In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.

Method A: Using Crtical values

Comparing the test statistic and tabulated critical value we have,

• If Test Statistic>Critical Value: Reject the null hypothesis.
• If Test Statistic≤Critical Value: Fail to reject the null hypothesis.

Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Method B: Using P-values

We can also come to an conclusion using the p-value,

Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Step 7- Interpret the Results

At last, we can conclude our experiment using method A or B.

Calculating test statistic

To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .

1. Z-statistics:

When population means and standard deviations are known.

• μ represents the population mean, 
• σ is the standard deviation
• and n is the size of the sample.

2. T-Statistics

T test is used when n<30,

t-statistic calculation is given by:

• t = t-score,
• x̄ = sample mean
• μ = population mean,
• s = standard deviation of the sample,
• n = sample size

3. Chi-Square Test

Chi-Square Test for Independence categorical Data (Non-normally distributed) using:

• i,j are the rows and columns index respectively.

Real life Hypothesis Testing example

Let’s examine hypothesis testing using two real life situations,

Case A: D oes a New Drug Affect Blood Pressure?

Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.

• Before Treatment: 120, 122, 118, 130, 125, 128, 115, 121, 123, 119
• After Treatment: 115, 120, 112, 128, 122, 125, 110, 117, 119, 114

Step 1 : Define the Hypothesis

• Null Hypothesis : (H 0 )The new drug has no effect on blood pressure.
• Alternate Hypothesis : (H 1 )The new drug has an effect on blood pressure.

Step 2: Define the Significance level

Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.

If the evidence suggests less than a 5% chance of observing the results due to random variation.

Step 3 : Compute the test statistic

Using paired T-test analyze the data to obtain a test statistic and a p-value.

The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.

t = m/(s/√n)

• m  = mean of the difference i.e X after, X before
• s  = standard deviation of the difference (d) i.e d i ​= X after, i ​− X before,
• n  = sample size,

then, m= -3.9, s= 1.8 and n= 10

we, calculate the , T-statistic = -9 based on the formula for paired t test

Step 4: Find the p-value

The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.

thus, p-value = 8.538051223166285e-06

Step 5: Result

• If the p-value is less than or equal to 0.05, the researchers reject the null hypothesis.
• If the p-value is greater than 0.05, they fail to reject the null hypothesis.

Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

Python Implementation of Hypothesis Testing

Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.

Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.

We will implement our first real life problem via python,

In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05. 

• The results suggest that the new drug, treatment, or intervention has a significant effect on lowering blood pressure.
• The negative T-statistic indicates that the mean blood pressure after treatment is significantly lower than the assumed population mean before treatment.

Case B : Cholesterol level in a population

Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.

Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.

Populations Mean = 200

Population Standard Deviation (σ): 5 mg/dL(given for this problem)

Step 1: Define the Hypothesis

• Null Hypothesis (H 0 ): The average cholesterol level in a population is 200 mg/dL.
• Alternate Hypothesis (H 1 ): The average cholesterol level in a population is different from 200 mg/dL.

As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.

Step 4: Result

Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL

Limitations of Hypothesis Testing

• Although a useful technique, hypothesis testing does not offer a comprehensive grasp of the topic being studied. Without fully reflecting the intricacy or whole context of the phenomena, it concentrates on certain hypotheses and statistical significance.
• The accuracy of hypothesis testing results is contingent on the quality of available data and the appropriateness of statistical methods used. Inaccurate data or poorly formulated hypotheses can lead to incorrect conclusions.
• Relying solely on hypothesis testing may cause analysts to overlook significant patterns or relationships in the data that are not captured by the specific hypotheses being tested. This limitation underscores the importance of complimenting hypothesis testing with other analytical approaches.

Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.

Frequently Asked Questions (FAQs)

1. what are the 3 types of hypothesis test.

There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.

2.What are the 4 components of hypothesis testing?

Null Hypothesis ( ): No effect or difference exists. Alternative Hypothesis ( ): An effect or difference exists. Significance Level ( ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.

3.What is hypothesis testing in ML?

Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.

4.What is the difference between Pytest and hypothesis in Python?

Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.

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Title: the platonic representation hypothesis.

Abstract: We argue that representations in AI models, particularly deep networks, are converging. First, we survey many examples of convergence in the literature: over time and across multiple domains, the ways by which different neural networks represent data are becoming more aligned. Next, we demonstrate convergence across data modalities: as vision models and language models get larger, they measure distance between datapoints in a more and more alike way. We hypothesize that this convergence is driving toward a shared statistical model of reality, akin to Plato's concept of an ideal reality. We term such a representation the platonic representation and discuss several possible selective pressures toward it. Finally, we discuss the implications of these trends, their limitations, and counterexamples to our analysis.

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Proactive green innovation and firm climate resilience: the nonlinear interaction effect of climate risk

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• Xinyi Gao 1 ,
• Siyuan Dong 2 ,
• Cheng Liu 3 &
• Hanying Wang 4  

Based on empirical analysis of 113 climate disasters affecting 3563 listed firms across 31 provinces in China from 2010 to 2022, as documented in the Emergency Events Database (EM-DAT), this study employs event study and multiple regression to explore the impact of proactive green innovation on firm climate resilience. By categorizing proactive green innovation into process and product innovation and climate resilience into short-term and long-term resilience, a proactive green innovation-firm climate resilience 2 × 2 matrix is constructed to provide innovative insights. This study reveals that proactive green innovation enhances firm climate resilience. Specifically, proactive green process innovation both enhances short-term and long-term climate resilience, while proactive green product innovation only enhances long-term rather than short-term climate resilience. Furthermore, climate disaster has inverted U-shaped interaction effect on the relationship between proactive green innovation and short-term climate resilience and U-shaped interaction effect on the relationship between proactive green innovation and long-term climate resilience. Additionally, this study also investigates the heterogeneous mechanisms of proactive green innovation enhancing short-term and long-term climate resilience based on network embeddedness theory and legitimacy theory.

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This work was supported by the Young Fund of National Social Science Foundation of China (Grant numbers: 19CGL023).

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All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Xinyi Gao and Cheng Liu. The first draft of the manuscript was written by Xinyi Gao. Writing—reviewing and editing were performed by Siyuan Dong and Hanying Wang. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Gao, X., Dong, S., Liu, C. et al. Proactive green innovation and firm climate resilience: the nonlinear interaction effect of climate risk. Environ Sci Pollut Res (2024). https://doi.org/10.1007/s11356-024-33576-4

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Received : 11 January 2024

Accepted : 30 April 2024

Published : 17 May 2024

DOI : https://doi.org/10.1007/s11356-024-33576-4

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