hypothesis testing in big data analytics

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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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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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Statistical Inference and Hypothesis Testing in Data Science Applications

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

Taught in English

Some content may not be translated

Instructor: Jem Corcoran

Financial aid available

5,438 already enrolled

(38 reviews)

Recommended experience

Intermediate level

Sequence in calculus up through Calculus II (preferably multivariate calculus) and some programming experience in R

What you'll learn

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.

Start Here!

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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A Guide on Data Analysis

14 hypothesis testing.

Error types:

Type I Error (False Positive):

• Reality: nope
• Diagnosis/Analysis: yes

Type II Error (False Negative):

• Reality: yes
• Diagnosis/Analysis: nope

Power: The probability of rejecting the null hypothesis when it is actually false

Always written in terms of the population parameter ( \(\beta\) ) not the estimator/estimate ( \(\hat{\beta}\) )

Sometimes, different disciplines prefer to use \(\beta\) (i.e., standardized coefficient), or \(\mathbf{b}\) (i.e., unstandardized coefficient)

\(\beta\) and \(\mathbf{b}\) are similar in interpretation; however, \(\beta\) is scale free. Hence, you can see the relative contribution of \(\beta\) to the dependent variable. On the other hand, \(\mathbf{b}\) can be more easily used in policy decisions.

\[ \beta_j = \mathbf{b} \frac{s_{x_j}}{s_y} \]

Assuming the null hypothesis is true, what is the (asymptotic) distribution of the estimator

\[ \begin{aligned} &H_0: \beta_j = 0 \\ &H_1: \beta_j \neq 0 \end{aligned} \]

then under the null, the OLS estimator has the following distribution

\[ A1-A3a, A5: \sqrt{n} \hat{\beta_j} \sim N(0,Avar(\sqrt{n}\hat{\beta}_j)) \]

• For the one-sided test, the null is a set of values, so now you choose the worst case single value that is hardest to prove and derive the distribution under the null

\[ \begin{aligned} &H_0: \beta_j\ge 0 \\ &H_1: \beta_j < 0 \end{aligned} \]

then the hardest null value to prove is \(H_0: \beta_j=0\) . Then under this specific null, the OLS estimator has the following asymptotic distribution

\[ A1-A3a, A5: \sqrt{n}\hat{\beta_j} \sim N(0,Avar(\sqrt{n}\hat{\beta}_j)) \]

14.1 Types of hypothesis testing

\(H_0 : \theta = \theta_0\)

\(H_1 : \theta \neq \theta_0\)

How far away / extreme \(\theta\) can be if our null hypothesis is true

Assume that our likelihood function for q is \(L(q) = q^{30}(1-q)^{70}\) Likelihood function

Log-Likelihood function

Figure from ( Fox 1997 )

typically, The likelihood ratio test (and Lagrange Multiplier (Score) ) performs better with small to moderate sample sizes, but the Wald test only requires one maximization (under the full model).

14.2 Wald test

\[ \begin{aligned} W &= (\hat{\theta}-\theta_0)'[cov(\hat{\theta})]^{-1}(\hat{\theta}-\theta_0) \\ W &\sim \chi_q^2 \end{aligned} \]

where \(cov(\hat{\theta})\) is given by the inverse Fisher Information matrix evaluated at \(\hat{\theta}\) and q is the rank of \(cov(\hat{\theta})\) , which is the number of non-redundant parameters in \(\theta\)

Alternatively,

\[ t_W=\frac{(\hat{\theta}-\theta_0)^2}{I(\theta_0)^{-1}} \sim \chi^2_{(v)} \]

where v is the degree of freedom.

Equivalently,

\[ s_W= \frac{\hat{\theta}-\theta_0}{\sqrt{I(\hat{\theta})^{-1}}} \sim Z \]

How far away in the distribution your sample estimate is from the hypothesized population parameter.

For a null value, what is the probability you would have obtained a realization “more extreme” or “worse” than the estimate you actually obtained?

Significance Level ( \(\alpha\) ) and Confidence Level ( \(1-\alpha\) )

• The significance level is the benchmark in which the probability is so low that we would have to reject the null
• The confidence level is the probability that sets the bounds on how far away the realization of the estimator would have to be to reject the null.

Test Statistics

• Standardized (transform) the estimator and null value to a test statistic that always has the same distribution
• Test Statistic for the OLS estimator for a single hypothesis

\[ T = \frac{\sqrt{n}(\hat{\beta}_j-\beta_{j0})}{\sqrt{n}SE(\hat{\beta_j})} \sim^a N(0,1) \]

\[ T = \frac{(\hat{\beta}_j-\beta_{j0})}{SE(\hat{\beta_j})} \sim^a N(0,1) \]

the test statistic is another random variable that is a function of the data and null hypothesis.

• T denotes the random variable test statistic
• t denotes the single realization of the test statistic

Evaluating Test Statistic: determine whether or not we reject or fail to reject the null hypothesis at a given significance / confidence level

Three equivalent ways

Critical Value

• Confidence Interval

For a given significance level, will determine the critical value \((c)\)

• One-sided: \(H_0: \beta_j \ge \beta_{j0}\)

\[ P(T<c|H_0)=\alpha \]

Reject the null if \(t<c\)

• One-sided: \(H_0: \beta_j \le \beta_{j0}\)

\[ P(T>c|H_0)=\alpha \]

Reject the null if \(t>c\)

• Two-sided: \(H_0: \beta_j \neq \beta_{j0}\)

\[ P(|T|>c|H_0)=\alpha \]

Reject the null if \(|t|>c\)

Calculate the probability that the test statistic was worse than the realization you have

\[ \text{p-value} = P(T<t|H_0) \]

\[ \text{p-value} = P(T>t|H_0) \]

\[ \text{p-value} = P(|T|<t|H_0) \]

reject the null if p-value \(< \alpha\)

Using the critical value associated with a null hypothesis and significance level, create an interval

\[ CI(\hat{\beta}_j)_{\alpha} = [\hat{\beta}_j-(c \times SE(\hat{\beta}_j)),\hat{\beta}_j+(c \times SE(\hat{\beta}_j))] \]

If the null set lies outside the interval then we reject the null.

• We are not testing whether the true population value is close to the estimate, we are testing that given a field true population value of the parameter, how like it is that we observed this estimate.
• Can be interpreted as we believe with \((1-\alpha)\times 100 \%\) probability that the confidence interval captures the true parameter value.

With stronger assumption (A1-A6), we could consider Finite Sample Properties

\[ T = \frac{\hat{\beta}_j-\beta_{j0}}{SE(\hat{\beta}_j)} \sim T(n-k) \]

• This above distributional derivation is strongly dependent on A4 and A5
• T has a student t-distribution because the numerator is normal and the denominator is \(\chi^2\) .
• Critical value and p-values will be calculated from the student t-distribution rather than the standard normal distribution.
• \(n \to \infty\) , \(T(n-k)\) is asymptotically standard normal.

Rule of thumb

if \(n-k>120\) : the critical values and p-values from the t-distribution are (almost) the same as the critical values and p-values from the standard normal distribution.

if \(n-k<120\)

• if (A1-A6) hold then the t-test is an exact finite distribution test
• if (A1-A3a, A5) hold, because the t-distribution is asymptotically normal, computing the critical values from a t-distribution is still a valid asymptotic test (i.e., not quite the right critical values and p0values, the difference goes away as \(n \to \infty\) )

14.2.1 Multiple Hypothesis

test multiple parameters as the same time

• \(H_0: \beta_1 = 0\ \& \ \beta_2 = 0\)
• \(H_0: \beta_1 = 1\ \& \ \beta_2 = 0\)

perform a series of simply hypothesis does not answer the question (joint distribution vs. two marginal distributions).

The test statistic is based on a restriction written in matrix form.

\[ y=\beta_0+x_1\beta_1 + x_2\beta_2 + x_3\beta_3 + \epsilon \]

Null hypothesis is \(H_0: \beta_1 = 0\) & \(\beta_2=0\) can be rewritten as \(H_0: \mathbf{R}\beta -\mathbf{q}=0\) where

• \(\mathbf{R}\) is a \(m \times k\) matrix where m is the number of restrictions and \(k\) is the number of parameters. \(\mathbf{q}\) is a \(k \times 1\) vector
• \(\mathbf{R}\) “picks up” the relevant parameters while \(\mathbf{q}\) is a the null value of the parameter

\[ \mathbf{R}= \left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{array} \right), \mathbf{q} = \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) \]

Test Statistic for OLS estimator for a multiple hypothesis

\[ F = \frac{(\mathbf{R\hat{\beta}-q})\hat{\Sigma}^{-1}(\mathbf{R\hat{\beta}-q})}{m} \sim^a F(m,n-k) \]

\(\hat{\Sigma}^{-1}\) is the estimator for the asymptotic variance-covariance matrix

• if A4 holds, both the homoskedastic and heteroskedastic versions produce valid estimator
• If A4 does not hold, only the heteroskedastic version produces valid estimators.

When \(m = 1\) , there is only a single restriction, then the \(F\) -statistic is the \(t\) -statistic squared.

\(F\) distribution is strictly positive, check F-Distribution for more details.

14.2.2 Linear Combination

Testing multiple parameters as the same time

\[ \begin{aligned} H_0&: \beta_1 -\beta_2 = 0 \\ H_0&: \beta_1 - \beta_2 > 0 \\ H_0&: \beta_1 - 2\times\beta_2 =0 \end{aligned} \]

Each is a single restriction on a function of the parameters.

Null hypothesis:

\[ H_0: \beta_1 -\beta_2 = 0 \]

can be rewritten as

\[ H_0: \mathbf{R}\beta -\mathbf{q}=0 \]

where \(\mathbf{R}\) =(0 1 -1 0 0) and \(\mathbf{q}=0\)

14.2.3 Estimate Difference in Coefficients

There is no package to estimate for the difference between two coefficients and its CI, but a simple function created by Katherine Zee can be used to calculate this difference. Some modifications might be needed if you don’t use standard lm model in R.

14.2.4 Application

14.2.5 nonlinear.

Suppose that we have q nonlinear functions of the parameters \[ \mathbf{h}(\theta) = \{ h_1 (\theta), ..., h_q (\theta)\}' \]

The,n, the Jacobian matrix ( \(\mathbf{H}(\theta)\) ), of rank q is

\[ \mathbf{H}_{q \times p}(\theta) = \left( \begin{array} {ccc} \frac{\partial h_1(\theta)}{\partial \theta_1} & ... & \frac{\partial h_1(\theta)}{\partial \theta_p} \\ . & . & . \\ \frac{\partial h_q(\theta)}{\partial \theta_1} & ... & \frac{\partial h_q(\theta)}{\partial \theta_p} \end{array} \right) \]

where the null hypothesis \(H_0: \mathbf{h} (\theta) = 0\) can be tested against the 2-sided alternative with the Wald statistic

\[ W = \frac{\mathbf{h(\hat{\theta})'\{H(\hat{\theta})[F(\hat{\theta})'F(\hat{\theta})]^{-1}H(\hat{\theta})'\}^{-1}h(\hat{\theta})}}{s^2q} \sim F_{q,n-p} \]

14.3 The likelihood ratio test

\[ t_{LR} = 2[l(\hat{\theta})-l(\theta_0)] \sim \chi^2_v \]

Compare the height of the log-likelihood of the sample estimate in relation to the height of log-likelihood of the hypothesized population parameter

This test considers a ratio of two maximizations,

\[ \begin{aligned} L_r &= \text{maximized value of the likelihood under $H_0$ (the reduced model)} \\ L_f &= \text{maximized value of the likelihood under $H_0 \cup H_a$ (the full model)} \end{aligned} \]

Then, the likelihood ratio is:

\[ \Lambda = \frac{L_r}{L_f} \]

which can’t exceed 1 (since \(L_f\) is always at least as large as \(L-r\) because \(L_r\) is the result of a maximization under a restricted set of the parameter values).

The likelihood ratio statistic is:

\[ \begin{aligned} -2ln(\Lambda) &= -2ln(L_r/L_f) = -2(l_r - l_f) \\ \lim_{n \to \infty}(-2ln(\Lambda)) &\sim \chi^2_v \end{aligned} \]

where \(v\) is the number of parameters in the full model minus the number of parameters in the reduced model.

If \(L_r\) is much smaller than \(L_f\) (the likelihood ratio exceeds \(\chi_{\alpha,v}^2\) ), then we reject he reduced model and accept the full model at \(\alpha \times 100 \%\) significance level

14.4 Lagrange Multiplier (Score)

\[ t_S= \frac{S(\theta_0)^2}{I(\theta_0)} \sim \chi^2_v \]

where \(v\) is the degree of freedom.

Compare the slope of the log-likelihood of the sample estimate in relation to the slope of the log-likelihood of the hypothesized population parameter

14.5 Two One-Sided Tests (TOST) Equivalence Testing

This is a good way to test whether your population effect size is within a range of practical interest (e.g., if the effect size is 0).

• > Machine Learning
• > Statistics

What is Hypothesis Testing? Types and Methods

• Soumyaa Rawat
• Jul 23, 2021

Hypothesis Testing  

Hypothesis testing is the act of testing a hypothesis or a supposition in relation to a statistical parameter. Analysts implement hypothesis testing in order to test if a hypothesis is plausible or not. 

In data science and statistics , hypothesis testing is an important step as it involves the verification of an assumption that could help develop a statistical parameter. For instance, a researcher establishes a hypothesis assuming that the average of all odd numbers is an even number. 

In order to find the plausibility of this hypothesis, the researcher will have to test the hypothesis using hypothesis testing methods. Unlike a hypothesis that is ‘supposed’ to stand true on the basis of little or no evidence, hypothesis testing is required to have plausible evidence in order to establish that a statistical hypothesis is true. 

Perhaps this is where statistics play an important role. A number of components are involved in this process. But before understanding the process involved in hypothesis testing in research methodology, we shall first understand the types of hypotheses that are involved in the process. Let us get started! 

Types of Hypotheses

In data sampling, different types of hypothesis are involved in finding whether the tested samples test positive for a hypothesis or not. In this segment, we shall discover the different types of hypotheses and understand the role they play in hypothesis testing.

Alternative Hypothesis

Alternative Hypothesis (H1) or the research hypothesis states that there is a relationship between two variables (where one variable affects the other). The alternative hypothesis is the main driving force for hypothesis testing. 

It implies that the two variables are related to each other and the relationship that exists between them is not due to chance or coincidence. 

When the process of hypothesis testing is carried out, the alternative hypothesis is the main subject of the testing process. The analyst intends to test the alternative hypothesis and verifies its plausibility.

Null Hypothesis

The Null Hypothesis (H0) aims to nullify the alternative hypothesis by implying that there exists no relation between two variables in statistics. It states that the effect of one variable on the other is solely due to chance and no empirical cause lies behind it. 

The null hypothesis is established alongside the alternative hypothesis and is recognized as important as the latter. In hypothesis testing, the null hypothesis has a major role to play as it influences the testing against the alternative hypothesis. 

(Must read: What is ANOVA test? )

Non-Directional Hypothesis

The Non-directional hypothesis states that the relation between two variables has no direction. 

Simply put, it asserts that there exists a relation between two variables, but does not recognize the direction of effect, whether variable A affects variable B or vice versa. 

Directional Hypothesis

The Directional hypothesis, on the other hand, asserts the direction of effect of the relationship that exists between two variables. 

Herein, the hypothesis clearly states that variable A affects variable B, or vice versa. 

Statistical Hypothesis

A statistical hypothesis is a hypothesis that can be verified to be plausible on the basis of statistics. 

By using data sampling and statistical knowledge, one can determine the plausibility of a statistical hypothesis and find out if it stands true or not. 

(Related blog: z-test vs t-test )

Performing Hypothesis Testing  

Now that we have understood the types of hypotheses and the role they play in hypothesis testing, let us now move on to understand the process in a better manner. 

In hypothesis testing, a researcher is first required to establish two hypotheses - alternative hypothesis and null hypothesis in order to begin with the procedure. 

To establish these two hypotheses, one is required to study data samples, find a plausible pattern among the samples, and pen down a statistical hypothesis that they wish to test. 

A random population of samples can be drawn, to begin with hypothesis testing. Among the two hypotheses, alternative and null, only one can be verified to be true. Perhaps the presence of both hypotheses is required to make the process successful. 

At the end of the hypothesis testing procedure, either of the hypotheses will be rejected and the other one will be supported. Even though one of the two hypotheses turns out to be true, no hypothesis can ever be verified 100%. 

(Read also: Types of data sampling techniques )

Therefore, a hypothesis can only be supported based on the statistical samples and verified data. Here is a step-by-step guide for hypothesis testing.

Establish the hypotheses

First things first, one is required to establish two hypotheses - alternative and null, that will set the foundation for hypothesis testing. 

These hypotheses initiate the testing process that involves the researcher working on data samples in order to either support the alternative hypothesis or the null hypothesis. 

Generate a testing plan

Once the hypotheses have been formulated, it is now time to generate a testing plan. A testing plan or an analysis plan involves the accumulation of data samples, determining which statistic is to be considered and laying out the sample size. 

All these factors are very important while one is working on hypothesis testing.

Analyze data samples

As soon as a testing plan is ready, it is time to move on to the analysis part. Analysis of data samples involves configuring statistical values of samples, drawing them together, and deriving a pattern out of these samples. 

While analyzing the data samples, a researcher needs to determine a set of things -

Significance Level - The level of significance in hypothesis testing indicates if a statistical result could have significance if the null hypothesis stands to be true.

Testing Method - The testing method involves a type of sampling-distribution and a test statistic that leads to hypothesis testing. There are a number of testing methods that can assist in the analysis of data samples. 

Test statistic - Test statistic is a numerical summary of a data set that can be used to perform hypothesis testing.

P-value - The P-value interpretation is the probability of finding a sample statistic to be as extreme as the test statistic, indicating the plausibility of the null hypothesis. 

Infer the results

The analysis of data samples leads to the inference of results that establishes whether the alternative hypothesis stands true or not. When the P-value is less than the significance level, the null hypothesis is rejected and the alternative hypothesis turns out to be plausible. 

Methods of Hypothesis Testing

As we have already looked into different aspects of hypothesis testing, we shall now look into the different methods of hypothesis testing. All in all, there are 2 most common types of hypothesis testing methods. They are as follows -

Frequentist Hypothesis Testing

The frequentist hypothesis or the traditional approach to hypothesis testing is a hypothesis testing method that aims on making assumptions by considering current data. 

The supposed truths and assumptions are based on the current data and a set of 2 hypotheses are formulated. A very popular subtype of the frequentist approach is the Null Hypothesis Significance Testing (NHST). 

The NHST approach (involving the null and alternative hypothesis) has been one of the most sought-after methods of hypothesis testing in the field of statistics ever since its inception in the mid-1950s. 

Bayesian Hypothesis Testing

A much unconventional and modern method of hypothesis testing, the Bayesian Hypothesis Testing claims to test a particular hypothesis in accordance with the past data samples, known as prior probability, and current data that lead to the plausibility of a hypothesis. 

The result obtained indicates the posterior probability of the hypothesis. In this method, the researcher relies on ‘prior probability and posterior probability’ to conduct hypothesis testing on hand. 

On the basis of this prior probability, the Bayesian approach tests a hypothesis to be true or false. The Bayes factor, a major component of this method, indicates the likelihood ratio among the null hypothesis and the alternative hypothesis. 

The Bayes factor is the indicator of the plausibility of either of the two hypotheses that are established for hypothesis testing.  

(Also read - Introduction to Bayesian Statistics ) 

To conclude, hypothesis testing, a way to verify the plausibility of a supposed assumption can be done through different methods - the Bayesian approach or the Frequentist approach. 

Although the Bayesian approach relies on the prior probability of data samples, the frequentist approach assumes without a probability. A number of elements involved in hypothesis testing are - significance level, p-level, test statistic, and method of hypothesis testing. 

(Also read: Introduction to probability distributions )

A significant way to determine whether a hypothesis stands true or not is to verify the data samples and identify the plausible hypothesis among the null hypothesis and alternative hypothesis. 

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Hypothesis Testing Steps & Examples

Table of Contents

What is a Hypothesis testing?

As per the definition from Oxford languages, a hypothesis is a supposition or proposed explanation made on the basis of limited evidence as a starting point for further investigation. As per the Dictionary page on Hypothesis , Hypothesis means a proposition or set of propositions, set forth as an explanation for the occurrence of some specified group of phenomena, either asserted merely as a provisional conjecture to guide investigation (working hypothesis) or accepted as highly probable in the light of established facts.

The hypothesis can be defined as the claim that can either be related to the truth about something that exists in the world, or, truth about something that’s needs to be established a fresh . In simple words, another word for the hypothesis is the “claim” . Until the claim is proven to be true, it is called the hypothesis. Once the claim is proved, it becomes the new truth or new knowledge about the thing. For example , let’s say that a claim is made that students studying for more than 6 hours a day gets more than 90% of marks in their examination. Now, this is just a claim or a hypothesis and not the truth in the real world. However, in order for the claim to become the truth for widespread adoption, it needs to be proved using pieces of evidence, e.g., data.  In order to reject this claim or otherwise, one needs to do some empirical analysis by gathering data samples and evaluating the claim. The process of gathering data and evaluating the claims or hypotheses with the goal to reject or otherwise (failing to reject) can be called as hypothesis testing . Note the wordings – “failing to reject”. It means that we don’t have enough evidence to reject the claim. Thus, until the time that new evidence comes up, the claim can be considered the truth. There are different techniques to test the hypothesis in order to reach the conclusion of whether the hypothesis can be used to represent the truth of the world.

One must note that the hypothesis testing never constitutes a proof that the hypothesis is absolute truth based on the observations. It only provides added support to consider the hypothesis as truth until the time that new evidences can against the hypotheses can be gathered. We can never be 100% sure about truth related to those hypotheses based on the hypothesis testing.

Simply speaking, hypothesis testing is a framework that can be used to assert whether the claim or the hypothesis made about a real-world/real-life event can be seen as the truth or otherwise based on the given data (evidences).

Hypothesis Testing Examples

Before we get ahead and start understanding more details about hypothesis and hypothesis testing steps, lets take a look at some  real-world examples  of how to think about hypothesis and hypothesis testing when dealing with real-world problems :

• Customers are churning because they ain’t getting response to their complaints or issues
• Customers are churning because there are other competitive services in the market which are providing these services at lower cost.
• Customers are churning because there are other competitive services which are providing more services at the same cost.
• It is claimed that a 500 gm sugar packet for a particular brand, say XYZA, contains sugar of less than 500 gm, say around 480gm.  Can this claim be taken as truth? How do we know that this claim is true? This is a hypothesis until proved.
• A group of doctors claims that quitting smoking increases lifespan. Can this claim be taken as new truth? The hypothesis is that quitting smoking results in an increase in lifespan.
• It is claimed that brisk walking for half an hour every day reverses diabetes. In order to accept this in your lifestyle, you may need evidence that supports this claim or hypothesis.
• It is claimed that doing Pranayama yoga for 30 minutes a day can help in easing stress by 50%. This can be termed as hypothesis and would require testing / validation for it to be established as a truth and recommended for widespread adoption.
• One common real-life example of hypothesis testing is election polling. In order to predict the outcome of an election, pollsters take a sample of the population and ask them who they plan to vote for. They then use hypothesis testing to assess whether their sample is representative of the population as a whole. If the results of the hypothesis test are significant, it means that the sample is representative and that the poll can be used to predict the outcome of the election. However, if the results are not significant, it means that the sample is not representative and that the poll should not be used to make predictions.
• Machine learning models make predictions based on the input data. Each of the machine learning model representing a function approximation can be taken as a hypothesis. All different models constitute what is called as hypothesis space .
• As part of a linear regression machine learning model , it is claimed that there is a relationship between the response variables and predictor variables? Can this hypothesis or claim be taken as truth? Let’s say, the hypothesis is that the housing price depends upon the average income of people already staying in the locality. How true is this hypothesis or claim? The relationship between response variable and each of the predictor variables can be evaluated using T-test and T-statistics .
• For linear regression model , one of the hypothesis is that there is no relationship between the response variable and any of the predictor variables. Thus, if b1, b2, b3 are three parameters, all of them is equal to 0. b1 = b2 = b3 = 0. This is where one performs F-test and use F-statistics to test this hypothesis.

You may note different hypotheses which are listed above. The next step would be validate some of these hypotheses. This is where data scientists will come into picture. One or more data scientists may be asked to work on different hypotheses. This would result in these data scientists looking for appropriate data related to the hypothesis they are working. This section will be detailed out in near future.

State the Hypothesis to begin Hypothesis Testing

The first step to hypothesis testing is defining or stating a hypothesis. Before the hypothesis can be tested, we need to formulate the hypothesis in terms of mathematical expressions. There are two important aspects to pay attention to, prior to the formulation of the hypothesis. The following represents different types of hypothesis that could be put to hypothesis testing:

• Claim made against the well-established fact : The case in which a fact is well-established, or accepted as truth or “knowledge” and a new claim is made about this well-established fact. For example , when you buy a packet of 500 gm of sugar, you assume that the packet does contain at the minimum 500 gm of sugar and not any less, based on the label of 500 gm on the packet. In this case, the fact is given or assumed to be the truth. A new claim can be made that the 500 gm sugar contains sugar weighing less than 500 gm. This claim needs to be tested before it is accepted as truth. Such cases could be considered for hypothesis testing if this is claimed that the assumption or the default state of being is not true. The claim to be established as new truth can be stated as “alternate hypothesis”. The opposite state can be stated as “null hypothesis”. Here the claim that the 500 gm packet consists of sugar less than 500 grams would be stated as alternate hypothesis. The opposite state which is the sugar packet consists 500 gm is null hypothesis.
• Claim to establish the new truth : The case in which there is some claim made about the reality that exists in the world (fact). For example , the fact that the housing price depends upon the average income of people already staying in the locality can be considered as a claim and not assumed to be true. Another example could be the claim that running 5 miles a day would result in a reduction of 10 kg of weight within a month. There could be varied such claims which when required to be proved as true have to go through hypothesis testing. The claim to be established as new truth can be stated as “alternate hypothesis”. The opposite state can be stated as “null hypothesis”. Running 5 miles a day would result in reduction of 10 kg within a month would be stated as alternate hypothesis.

Based on the above considerations, the following hypothesis can be stated for doing hypothesis testing.

• The packet of 500 gm of sugar contains sugar of weight less than 500 gm. (Claim made against the established fact). This is a new knowledge which requires hypothesis testing to get established and acted upon.
• The housing price depends upon the average income of the people staying in the locality. This is a new knowledge which requires hypothesis testing to get established and acted upon.
• Running 5 miles a day results in a reduction of 10 kg of weight within a month. This is a new knowledge which requires hypothesis testing to get established for widespread adoption.

Formulate Null & Alternate Hypothesis as Next Step

Once the hypothesis is defined or stated, the next step is to formulate the null and alternate hypothesis in order to begin hypothesis testing as described above.

What is a null hypothesis?

In the case where the given statement is a well-established fact or default state of being in the real world, one can call it a null hypothesis (in the simpler word, nothing new). Well-established facts don’t need any hypothesis testing and hence can be called the null hypothesis. In cases, when there are any new claims made which is not well established in the real world, the null hypothesis can be thought of as the default state or opposite state of that claim. For example , in the previous section, the claim or hypothesis is made that the students studying for more than 6 hours a day gets more than 90% of marks in their examination. The null hypothesis, in this case, will be that the claim is not true or real. The null hypothesis can be stated that there is no relationship or association between the students reading more than 6 hours a day and they getting 90% of the marks. Any occurrence is only a chance occurrence. Another example of hypothesis is when somebody is alleged that they have performed a crime.

Null hypothesis is denoted by letter H with 0, e.g., [latex]H_0[/latex]

What is an alternate hypothesis?

When the given statement is a claim (unexpected event in the real world) and not yet proven, one can call/formulate it as an alternate hypothesis and accordingly define a null hypothesis which is the opposite state of the hypothesis. The alternate hypothesis is a new knowledge or truth that needs to be established. In simple words, the hypothesis or claim that needs to be tested against reality in the real world can be termed the alternate hypothesis. In order to reach a conclusion that the claim (alternate hypothesis) can be considered the new knowledge or truth (based on the available evidence), it would be important to reject the null hypothesis. It should be noted that null and alternate hypotheses are mutually exclusive and at the same time asymmetric. In the example given in the previous section, the claim that the students studying for more than 6 hours get more than 90% of marks can be termed as the alternate hypothesis.

Alternate hypothesis is denoted with H subscript a, e.g., [latex]H_a[/latex]

Once the hypothesis is formulated as null([latex]H_0[/latex]) and alternate hypothesis ([latex]H_a[/latex]), there are two possible outcomes that can happen from hypothesis testing. These outcomes are the following:

• Reject the null hypothesis : There is enough evidence based on which one can reject the null hypothesis. Let’s understand this with the help of an example provided earlier in this section. The null hypothesis is that there is no relationship between the students studying more than 6 hours a day and getting more than 90% marks. In a sample of 30 students studying more than 6 hours a day, it was found that they scored 91% marks. Given that the null hypothesis is true, this kind of hypothesis testing result will be highly unlikely. This kind of result can’t happen by chance. That would mean that the claim can be taken as the new truth or new knowledge in the real world. One can go and take further samples of 30 students to perform some more testing to validate the hypothesis. If similar results show up with other tests, it can be said with very high confidence that there is enough evidence to reject the null hypothesis that there is no relationship between the students studying more than 6 hours a day and getting more than 90% marks. In such cases, one can go to accept the claim as new truth that the students studying more than 6 hours a day get more than 90% marks. The hypothesis can be considered the new truth until the time that new tests provide evidence against this claim.
• Fail to reject the null hypothesis : There is not enough evidence-based on which one can reject the null hypothesis (well-established fact or reality). Thus, one would fail to reject the null hypothesis. In a sample of 30 students studying more than 6 hours a day, the students were found to score 75%. Given that the null hypothesis is true, this kind of result is fairly likely or expected. With the given sample, one can’t reject the null hypothesis that there is no relationship between the students studying more than 6 hours a day and getting more than 90% marks.

Examples of formulating the null and alternate hypothesis

The following are some examples of the null and alternate hypothesis.

Hypothesis Testing Steps

Here is the diagram which represents the workflow of Hypothesis Testing.

Figure 1. Hypothesis Testing Steps

Based on the above, the following are some of the  steps to be taken when doing hypothesis testing:

• State the hypothesis : First and foremost, the hypothesis needs to be stated. The hypothesis could either be the statement that is assumed to be true or the claim which is made to be true.
• Formulate the hypothesis : This step requires one to identify the Null and Alternate hypotheses or in simple words, formulate the hypothesis. Take an example of the canned sauce weighing 500 gm as the Null Hypothesis.
• Set the criteria for a decision : Identify test statistics that could be used to assess the Null Hypothesis. The test statistics with the above example would be the average weight of the sugar packet, and t-statistics would be used to determine the P-value. For different kinds of problems, different kinds of statistics including Z-statistics, T-statistics, F-statistics, etc can be used.
• Identify the level of significance (alpha) : Before starting the hypothesis testing, one would be required to set the significance level (also called as  alpha ) which represents the value for which a P-value less than or equal to  alpha  is considered statistically significant. Typical values of  alpha  are 0.1, 0.05, and 0.01. In case the P-value is evaluated as statistically significant, the null hypothesis is rejected. In case, the P-value is more than the  alpha  value, the null hypothesis is failed to be rejected.
• Compute the test statistics : Next step is to calculate the test statistics (z-test, t-test, f-test, etc) to determine the P-value. If the sample size is more than 30, it is recommended to use z-statistics. Otherwise, t-statistics could be used. In the current example where 20 packets of canned sauce is selected for hypothesis testing, t-statistics will be calculated for the mean value of 505 gm (sample mean). The t-statistics would then be calculated as the difference of 505 gm (sample mean) and the population means (500 gm) divided by the sample standard deviation divided by the square root of sample size (20).
• Calculate the P-value of the test statistics : Once the test statistics have been calculated, find the P-value using either of t-table or a z-table. P-value is the probability of obtaining a test statistic (t-score or z-score) equal to or more extreme than the result obtained from the sample data, given that the null hypothesis H0 is true.
• Compare P-value with the level of significance : The significance level is set as the allowable range within which if the value appears, one will be failed to reject the Null Hypothesis. This region is also called as Non-rejection region . The value of alpha is compared with the p-value. If the p-value is less than the significance level, the test is statistically significant and hence, the null hypothesis will be rejected.

P-Value: Key to Statistical Hypothesis Testing

Once you formulate the hypotheses, there is the need to test those hypotheses. Meaning, say that the null hypothesis is stated as the statement that housing price does not depend upon the average income of people staying in the locality, it would be required to be tested by taking samples of housing prices and, based on the test results, this Null hypothesis could either be rejected or failed to be rejected . In hypothesis testing, the following two are the outcomes:

• Reject the Null hypothesis
• Fail to Reject the Null hypothesis

Take the above example of the sugar packet weighing 500 gm. The Null hypothesis is set as the statement that the sugar packet weighs 500 gm. After taking a sample of 20 sugar packets and testing/taking its weight, it was found that the average weight of the sugar packets came to 495 gm. The test statistics (t-statistics) were calculated for this sample and the P-value was determined. Let’s say the P-value was found to be 15%. Assuming that the level of significance is selected to be 5%, the test statistic is not statistically significant (P-value > 5%) and thus, the null hypothesis fails to get rejected. Thus, one could safely conclude that the sugar packet does weigh 500 gm. However, if the average weight of canned sauce would have found to be 465 gm, this is way beyond/away from the mean value of 500 gm and one could have ended up rejecting the Null Hypothesis based on the P-value .

Hypothesis Testing for Problem Analysis & Solution Implementation

Hypothesis testing can be applied in both problem analysis and solution implementation. The following represents method on how you can apply hypothesis testing technique for both problem and solution space:

• Problem Analysis : Hypothesis testing is a systematic way to validate assumptions or educated guesses during problem analysis. It allows for a structured investigation into the nature of a problem and its potential root causes. In this process, a null hypothesis and an alternative hypothesis are usually defined. The null hypothesis generally asserts that no significant change or effect exists, while the alternative hypothesis posits the opposite. Through controlled experiments, data collection, or statistical analysis, these hypotheses are then tested to determine their validity. For example, if a software company notices a sudden increase in user churn rate, they might hypothesize that the recent update to their application is the root cause. The null hypothesis could be that the update has no effect on churn rate, while the alternative hypothesis would assert that the update significantly impacts the churn rate. By analyzing user behavior and feedback before and after the update, and perhaps running A/B tests where one user group has the update and another doesn’t, the company can test these hypotheses. If the alternative hypothesis is confirmed, the company can then focus on identifying specific issues in the update that may be causing the increased churn, thereby moving closer to a solution.
• Solution Implementation : Hypothesis testing can also be a valuable tool during the solution implementation phase, serving as a method to evaluate the effectiveness of proposed remedies. By setting up a specific hypothesis about the expected outcome of a solution, organizations can create targeted metrics and KPIs to measure success. For example, if a retail business is facing low customer retention rates, they might implement a loyalty program as a solution. The hypothesis could be that introducing a loyalty program will increase customer retention by at least 15% within six months. The null hypothesis would state that the loyalty program has no significant effect on retention rates. To test this, the company can compare retention metrics from before and after the program’s implementation, possibly even setting up control groups for more robust analysis. By applying statistical tests to this data, the company can determine whether their hypothesis is confirmed or refuted, thereby gauging the effectiveness of their solution and making data-driven decisions for future actions.
• Tests of Significance
• Hypothesis testing for the Mean
• z-statistics vs t-statistics (Khan Academy)

Hypothesis testing quiz

The claim that needs to be established is set as ____________, the outcome of hypothesis testing is _________.

Please select 2 correct answers

P-value is defined as the probability of obtaining the result as extreme given the null hypothesis is true

There is a claim that doing pranayama yoga results in reversing diabetes. which of the following is true about null hypothesis.

In this post, you learned about hypothesis testing and related nuances such as the null and alternate hypothesis formulation techniques, ways to go about doing hypothesis testing etc. In data science, one of the reasons why one needs to understand the concepts of hypothesis testing is the need to verify the relationship between the dependent (response) and independent (predictor) variables. One would, thus, need to understand the related concepts such as hypothesis formulation into null and alternate hypothesis, level of significance, test statistics calculation, P-value, etc. Given that the relationship between dependent and independent variables is a sort of hypothesis or claim , the null hypothesis could be set as the scenario where there is no relationship between dependent and independent variables.

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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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Ladislaus von Bortkiewicz Chair of Statistics, C.A.S.E. Center for Applied Statistics & Economics, Humboldt-Universität zu Berlin, Berlin, Germany

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Institute of Statistics, National Chiao Tung University, Hsinchu, Taiwan

School of statistics, university of minnesota, minneapolis, usa.

• Offers a valuable guide to a broad range of big data analytics with statistics in cross-disciplinary applications
• Shows how to handle high-dimensional problems in big data analytics
• Offers software-hardware co-designs for big data analytics

Part of the book series: Springer Handbooks of Computational Statistics (SHCS)

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Table of contents (21 chapters)

Front matter, statistics, statisticians, and the internet of things.

• John M. Jordan, Dennis K. J. Lin

Cognitive Data Analysis for Big Data

• Jing Shyr, Jane Chu, Mike Woods

Methodology

Statistical leveraging methods in big data.

• Xinlian Zhang, Rui Xie, Ping Ma

Scattered Data and Aggregated Inference

• Xiaoming Huo, Cheng Huang, Xuelei Sherry Ni

Nonparametric Methods for Big Data Analytics

• Hao Helen Zhang

Finding Patterns in Time Series

• James E. Gentle, Seunghye J. Wilson

Variational Bayes for Hierarchical Mixture Models

• Muting Wan, James G. Booth, Martin T. Wells

Hypothesis Testing for High-Dimensional Data

• Wei Biao Wu, Zhipeng Lou, Yuefeng Han

High-Dimensional Classification

Analysis of high-dimensional regression models using orthogonal greedy algorithms.

• Hsiang-Ling Hsu, Ching-Kang Ing, Tze Leung Lai

Semi-supervised Smoothing for Large Data Problems

• Mark Vere Culp, Kenneth Joseph Ryan, George Michailidis

Inverse Modeling: A Strategy to Cope with Non-linearity

• Qian Lin, Yang Li, Jun S. Liu

Sufficient Dimension Reduction for Tensor Data

• Yiwen Liu, Xin Xing, Wenxuan Zhong

Compressive Sensing and Sparse Coding

• Kevin Chen, H. T. Kung

Bridging Density Functional Theory and Big Data Analytics with Applications

• Chien-Chang Chen, Hung-Hui Juan, Meng-Yuan Tsai, Henry Horng-Shing Lu

Q3-D3-LSA: D3.js and Generalized Vector Space Models for Statistical Computing

• Lukas Borke, Wolfgang K. Härdle
• Computational Statistics
• Data Analytics
• High-Dimensional Data Analysis
• Software-Hardware Co-Designs

About this book

Editors and affiliations, ladislaus von bortkiewicz chair of statistics, c.a.s.e. center for applied statistics & economics, humboldt-universität zu berlin, berlin, germany.

Wolfgang Karl Härdle

Henry Horng-Shing Lu

Xiaotong Shen

About the editors

Wolfgang Karl Härdle  is Ladislaus von Bortkievicz Professor of Statistics at the Humboldt University of Berlin and director of C.A.S.E. (Center for Applied Statistics and Economics), director of the Collaborative Research Center 649 "Economic Risk" and also of the IRTG 1792 "High Dimensional Nonstationary Time Series". He teaches quantitative finance and semiparametric statistics. Professor Härdle's research focuses on dynamic factor models, multivariate statistics in finance and computational statistics. He is an elected member of the International Statistical Institute (ISI) and advisor to the Guanghua School of Management, Peking University, China.

Henry Horng-Shing Lu  is Professor at the Institute of Statistics of the National Chiao Tung University, Taiwan and serves as the Vice President of Academic Affairs. He received his Ph.D. in Statistics from Cornell University, NY in 1994. He is an elected member of the International Statistical Institute (ISI). His research interests include statistics, applications and big data analytics. Professor Lu analyzes different types of data by developing statistical methodologies for machine learning with the power of statistical inference and computation algorithms. His findings were published in a wide spectrum of journals and conference papers. He also co-edited the Handbook of Statistical Bioinformatics , published by Springer in 2011.

Xiaotong Shen  is John Black Johnston Distinguished Professor at the School of Statistics of the University of Minnesota, MN. He received his Ph.D. in Statistics from the University of Chicago, IL in 1991. He is Fellow of the American Statistical Association (ASA), the Institute of Mathematical Statistics (IMS), and the American Association for the Advancement of Science (AAAS) as well as an elected member of the International Statistical Institute (ISI). Professor Shen’s areas of interest include machine learning anddata mining, likelihood-based inference, semiparametric and nonparametric models, model selection and averaging. His current research efforts are mainly devoted to the further development of structured learning as well as high-dimensional/high-order analysis. The targeted application areas are biomedical sciences and engineering.

Bibliographic Information

Book Title : Handbook of Big Data Analytics

Editors : Wolfgang Karl Härdle, Henry Horng-Shing Lu, Xiaotong Shen

Series Title : Springer Handbooks of Computational Statistics

DOI : https://doi.org/10.1007/978-3-319-18284-1

Publisher : Springer Cham

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : Springer International Publishing AG, part of Springer Nature 2018

Hardcover ISBN : 978-3-319-18283-4 Published: 01 August 2018

Softcover ISBN : 978-3-030-13238-5 Published: 10 December 2019

eBook ISBN : 978-3-319-18284-1 Published: 20 July 2018

Series ISSN : 2197-9790

Series E-ISSN : 2197-9804

Edition Number : 1

Number of Pages : VIII, 538

Number of Illustrations : 38 b/w illustrations, 109 illustrations in colour

Topics : Statistics and Computing/Statistics Programs , Data Mining and Knowledge Discovery , Mathematical and Computational Engineering , Statistical Theory and Methods , Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences

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Tutorial Playlist

Statistics tutorial, everything you need to know about the probability density function in statistics, the best guide to understand central limit theorem, an in-depth guide to measures of central tendency : mean, median and mode, the ultimate guide to understand conditional probability.

A Comprehensive Look at Percentile in Statistics

The Best Guide to Understand Bayes Theorem

Everything you need to know about the normal distribution, an in-depth explanation of cumulative distribution function, a complete guide to chi-square test, what is hypothesis testing in statistics types and examples, understanding the fundamentals of arithmetic and geometric progression, the definitive guide to understand spearman’s rank correlation, a comprehensive guide to understand mean squared error, all you need to know about the empirical rule in statistics, the complete guide to skewness and kurtosis, a holistic look at bernoulli distribution.

All You Need to Know About Bias in Statistics

A Complete Guide to Get a Grasp of Time Series Analysis

The Key Differences Between Z-Test Vs. T-Test

The Complete Guide to Understand Pearson's Correlation

A complete guide on the types of statistical studies, everything you need to know about poisson distribution, your best guide to understand correlation vs. regression, the most comprehensive guide for beginners on what is correlation, what is hypothesis testing in statistics types and examples.

Lesson 10 of 24 By Avijeet Biswal

Table of Contents

In today’s data-driven world , decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis & hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

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What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life - 

• A teacher assumes that 60% of his college's students come from lower-middle-class families.
• A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

• Here, x̅ is the sample mean,
• μ0 is the population mean,
• σ is the standard deviation,
• n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

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Null Hypothesis and Alternate Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average. 

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

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Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine that their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps of Hypothesis Testing

Hypothesis testing is a statistical method to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. Here’s a breakdown of the typical steps involved in hypothesis testing:

Formulate Hypotheses

• Null Hypothesis (H0): This hypothesis states that there is no effect or difference, and it is the hypothesis you attempt to reject with your test.
• Alternative Hypothesis (H1 or Ha): This hypothesis is what you might believe to be true or hope to prove true. It is usually considered the opposite of the null hypothesis.

Choose the Significance Level (α)

The significance level, often denoted by alpha (α), is the probability of rejecting the null hypothesis when it is true. Common choices for α are 0.05 (5%), 0.01 (1%), and 0.10 (10%).

Select the Appropriate Test

Choose a statistical test based on the type of data and the hypothesis. Common tests include t-tests, chi-square tests, ANOVA, and regression analysis . The selection depends on data type, distribution, sample size, and whether the hypothesis is one-tailed or two-tailed.

Collect Data

Gather the data that will be analyzed in the test. This data should be representative of the population to infer conclusions accurately.

Calculate the Test Statistic

Based on the collected data and the chosen test, calculate a test statistic that reflects how much the observed data deviates from the null hypothesis.

Determine the p-value

The p-value is the probability of observing test results at least as extreme as the results observed, assuming the null hypothesis is correct. It helps determine the strength of the evidence against the null hypothesis.

Make a Decision

Compare the p-value to the chosen significance level:

• If the p-value ≤ α: Reject the null hypothesis, suggesting sufficient evidence in the data supports the alternative hypothesis.
• If the p-value > α: Do not reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.

Report the Results

Present the findings from the hypothesis test, including the test statistic, p-value, and the conclusion about the hypotheses.

Perform Post-hoc Analysis (if necessary)

Depending on the results and the study design, further analysis may be needed to explore the data more deeply or to address multiple comparisons if several hypotheses were tested simultaneously.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

Chi-Square 

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

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Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

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Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

• The null hypothesis is (H0 <= 90) or less change.
• A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true]. 

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

Level of Significance

The alpha value is a criterion for determining whether a test statistic is statistically significant. In a statistical test, Alpha represents an acceptable probability of a Type I error. Because alpha is a probability, it can be anywhere between 0 and 1. In practice, the most commonly used alpha values are 0.01, 0.05, and 0.1, which represent a 1%, 5%, and 10% chance of a Type I error, respectively (i.e. rejecting the null hypothesis when it is in fact correct).

A p-value is a metric that expresses the likelihood that an observed difference could have occurred by chance. As the p-value decreases the statistical significance of the observed difference increases. If the p-value is too low, you reject the null hypothesis.

Here you have taken an example in which you are trying to test whether the new advertising campaign has increased the product's sales. The p-value is the likelihood that the null hypothesis, which states that there is no change in the sales due to the new advertising campaign, is true. If the p-value is .30, then there is a 30% chance that there is no increase or decrease in the product's sales.  If the p-value is 0.03, then there is a 3% probability that there is no increase or decrease in the sales value due to the new advertising campaign. As you can see, the lower the p-value, the chances of the alternate hypothesis being true increases, which means that the new advertising campaign causes an increase or decrease in sales.

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Why Is Hypothesis Testing Important in Research Methodology?

Hypothesis testing is crucial in research methodology for several reasons:

• Provides evidence-based conclusions: It allows researchers to make objective conclusions based on empirical data, providing evidence to support or refute their research hypotheses.
• Supports decision-making: It helps make informed decisions, such as accepting or rejecting a new treatment, implementing policy changes, or adopting new practices.
• Adds rigor and validity: It adds scientific rigor to research using statistical methods to analyze data, ensuring that conclusions are based on sound statistical evidence.
• Contributes to the advancement of knowledge: By testing hypotheses, researchers contribute to the growth of knowledge in their respective fields by confirming existing theories or discovering new patterns and relationships.

When Did Hypothesis Testing Begin?

Hypothesis testing as a formalized process began in the early 20th century, primarily through the work of statisticians such as Ronald A. Fisher, Jerzy Neyman, and Egon Pearson. The development of hypothesis testing is closely tied to the evolution of statistical methods during this period.

• Ronald A. Fisher (1920s): Fisher was one of the key figures in developing the foundation for modern statistical science. In the 1920s, he introduced the concept of the null hypothesis in his book "Statistical Methods for Research Workers" (1925). Fisher also developed significance testing to examine the likelihood of observing the collected data if the null hypothesis were true. He introduced p-values to determine the significance of the observed results.
• Neyman-Pearson Framework (1930s): Jerzy Neyman and Egon Pearson built on Fisher’s work and formalized the process of hypothesis testing even further. In the 1930s, they introduced the concepts of Type I and Type II errors and developed a decision-making framework widely used in hypothesis testing today. Their approach emphasized the balance between these errors and introduced the concepts of the power of a test and the alternative hypothesis.

The dialogue between Fisher's and Neyman-Pearson's approaches shaped the methods and philosophy of statistical hypothesis testing used today. Fisher emphasized the evidential interpretation of the p-value. At the same time, Neyman and Pearson advocated for a decision-theoretical approach in which hypotheses are either accepted or rejected based on pre-determined significance levels and power considerations.

The application and methodology of hypothesis testing have since become a cornerstone of statistical analysis across various scientific disciplines, marking a significant statistical development.

Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

• It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
• Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
• Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
• Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

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After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

If you are interested in statistics of data science and skills needed for such a career, you ought to explore the Post Graduate Program in Data Science.

If you have any questions regarding this ‘Hypothesis Testing In Statistics’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning!

1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is H0 and H1 in statistics?

In statistics, H0​ and H1​ represent the null and alternative hypotheses. The null hypothesis, H0​, is the default assumption that no effect or difference exists between groups or conditions. The alternative hypothesis, H1​, is the competing claim suggesting an effect or a difference. Statistical tests determine whether to reject the null hypothesis in favor of the alternative hypothesis based on the data.

3. What is a simple hypothesis with an example?

A simple hypothesis is a specific statement predicting a single relationship between two variables. It posits a direct and uncomplicated outcome. For example, a simple hypothesis might state, "Increased sunlight exposure increases the growth rate of sunflowers." Here, the hypothesis suggests a direct relationship between the amount of sunlight (independent variable) and the growth rate of sunflowers (dependent variable), with no additional variables considered.

4. What are the 2 types of hypothesis testing?

• One-tailed (or one-sided) test: Tests for the significance of an effect in only one direction, either positive or negative.
• Two-tailed (or two-sided) test: Tests for the significance of an effect in both directions, allowing for the possibility of a positive or negative effect.

The choice between one-tailed and two-tailed tests depends on the specific research question and the directionality of the expected effect.

5. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

• Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
• Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
• Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

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About the author.

Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

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