hypothesis test 2 variables

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• This table is designed to help you choose an appropriate statistical test for data with one dependent variable .
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Statistics By Jim

Making statistics intuitive

Statistical Hypothesis Testing Overview

By Jim Frost 59 Comments

In this blog post, I explain why you need to use statistical hypothesis testing and help you navigate the essential terminology. Hypothesis testing is a crucial procedure to perform when you want to make inferences about a population using a random sample. These inferences include estimating population properties such as the mean, differences between means, proportions, and the relationships between variables.

This post provides an overview of statistical hypothesis testing. If you need to perform hypothesis tests, consider getting my book, Hypothesis Testing: An Intuitive Guide .

Why You Should Perform Statistical Hypothesis Testing

Hypothesis testing is a form of inferential statistics that allows us to draw conclusions about an entire population based on a representative sample. You gain tremendous benefits by working with a sample. In most cases, it is simply impossible to observe the entire population to understand its properties. The only alternative is to collect a random sample and then use statistics to analyze it.

While samples are much more practical and less expensive to work with, there are trade-offs. When you estimate the properties of a population from a sample, the sample statistics are unlikely to equal the actual population value exactly.  For instance, your sample mean is unlikely to equal the population mean. The difference between the sample statistic and the population value is the sample error.

Differences that researchers observe in samples might be due to sampling error rather than representing a true effect at the population level. If sampling error causes the observed difference, the next time someone performs the same experiment the results might be different. Hypothesis testing incorporates estimates of the sampling error to help you make the correct decision. Learn more about Sampling Error .

For example, if you are studying the proportion of defects produced by two manufacturing methods, any difference you observe between the two sample proportions might be sample error rather than a true difference. If the difference does not exist at the population level, you won’t obtain the benefits that you expect based on the sample statistics. That can be a costly mistake!

Let’s cover some basic hypothesis testing terms that you need to know.

Background information : Difference between Descriptive and Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics

Hypothesis Testing

Hypothesis testing is a statistical analysis that uses sample data to assess two mutually exclusive theories about the properties of a population. Statisticians call these theories the null hypothesis and the alternative hypothesis. A hypothesis test assesses your sample statistic and factors in an estimate of the sample error to determine which hypothesis the data support.

When you can reject the null hypothesis, the results are statistically significant, and your data support the theory that an effect exists at the population level.

The effect is the difference between the population value and the null hypothesis value. The effect is also known as population effect or the difference. For example, the mean difference between the health outcome for a treatment group and a control group is the effect.

Typically, you do not know the size of the actual effect. However, you can use a hypothesis test to help you determine whether an effect exists and to estimate its size. Hypothesis tests convert your sample effect into a test statistic, which it evaluates for statistical significance. Learn more about Test Statistics .

An effect can be statistically significant, but that doesn’t necessarily indicate that it is important in a real-world, practical sense. For more information, read my post about Statistical vs. Practical Significance .

Null Hypothesis

The null hypothesis is one of two mutually exclusive theories about the properties of the population in hypothesis testing. Typically, the null hypothesis states that there is no effect (i.e., the effect size equals zero). The null is often signified by H 0 .

In all hypothesis testing, the researchers are testing an effect of some sort. The effect can be the effectiveness of a new vaccination, the durability of a new product, the proportion of defect in a manufacturing process, and so on. There is some benefit or difference that the researchers hope to identify.

However, it’s possible that there is no effect or no difference between the experimental groups. In statistics, we call this lack of an effect the null hypothesis. Therefore, if you can reject the null, you can favor the alternative hypothesis, which states that the effect exists (doesn’t equal zero) at the population level.

You can think of the null as the default theory that requires sufficiently strong evidence against in order to reject it.

For example, in a 2-sample t-test, the null often states that the difference between the two means equals zero.

When you can reject the null hypothesis, your results are statistically significant. Learn more about Statistical Significance: Definition & Meaning .

Related post : Understanding the Null Hypothesis in More Detail

Alternative Hypothesis

The alternative hypothesis is the other theory about the properties of the population in hypothesis testing. Typically, the alternative hypothesis states that a population parameter does not equal the null hypothesis value. In other words, there is a non-zero effect. If your sample contains sufficient evidence, you can reject the null and favor the alternative hypothesis. The alternative is often identified with H 1 or H A .

For example, in a 2-sample t-test, the alternative often states that the difference between the two means does not equal zero.

You can specify either a one- or two-tailed alternative hypothesis:

If you perform a two-tailed hypothesis test, the alternative states that the population parameter does not equal the null value. For example, when the alternative hypothesis is H A : μ ≠ 0, the test can detect differences both greater than and less than the null value.

A one-tailed alternative has more power to detect an effect but it can test for a difference in only one direction. For example, H A : μ > 0 can only test for differences that are greater than zero.

Related posts : Understanding T-tests and One-Tailed and Two-Tailed Hypothesis Tests Explained

P-values are the probability that you would obtain the effect observed in your sample, or larger, if the null hypothesis is correct. In simpler terms, p-values tell you how strongly your sample data contradict the null. Lower p-values represent stronger evidence against the null. You use P-values in conjunction with the significance level to determine whether your data favor the null or alternative hypothesis.

Related post : Interpreting P-values Correctly

Significance Level (Alpha)

For instance, a significance level of 0.05 signifies a 5% risk of deciding that an effect exists when it does not exist.

Use p-values and significance levels together to help you determine which hypothesis the data support. If the p-value is less than your significance level, you can reject the null and conclude that the effect is statistically significant. In other words, the evidence in your sample is strong enough to be able to reject the null hypothesis at the population level.

Related posts : Graphical Approach to Significance Levels and P-values and Conceptual Approach to Understanding Significance Levels

Types of Errors in Hypothesis Testing

Statistical hypothesis tests are not 100% accurate because they use a random sample to draw conclusions about entire populations. There are two types of errors related to drawing an incorrect conclusion.

• False positives: You reject a null that is true. Statisticians call this a Type I error . The Type I error rate equals your significance level or alpha (α).
• False negatives: You fail to reject a null that is false. Statisticians call this a Type II error. Generally, you do not know the Type II error rate. However, it is a larger risk when you have a small sample size , noisy data, or a small effect size. The type II error rate is also known as beta (β).

Statistical power is the probability that a hypothesis test correctly infers that a sample effect exists in the population. In other words, the test correctly rejects a false null hypothesis. Consequently, power is inversely related to a Type II error. Power = 1 – β. Learn more about Power in Statistics .

Related posts : Types of Errors in Hypothesis Testing and Estimating a Good Sample Size for Your Study Using Power Analysis

Which Type of Hypothesis Test is Right for You?

There are many different types of procedures you can use. The correct choice depends on your research goals and the data you collect. Do you need to understand the mean or the differences between means? Or, perhaps you need to assess proportions. You can even use hypothesis testing to determine whether the relationships between variables are statistically significant.

To choose the proper statistical procedure, you’ll need to assess your study objectives and collect the correct type of data . This background research is necessary before you begin a study.

Related Post : Hypothesis Tests for Continuous, Binary, and Count Data

Statistical tests are crucial when you want to use sample data to make conclusions about a population because these tests account for sample error. Using significance levels and p-values to determine when to reject the null hypothesis improves the probability that you will draw the correct conclusion.

To see an alternative approach to these traditional hypothesis testing methods, learn about bootstrapping in statistics !

If you want to see examples of hypothesis testing in action, I recommend the following posts that I have written:

• How Effective Are Flu Shots? This example shows how you can use statistics to test proportions.
• Fatality Rates in Star Trek . This example shows how to use hypothesis testing with categorical data.
• Busting Myths About the Battle of the Sexes . A fun example based on a Mythbusters episode that assess continuous data using several different tests.
• Are Yawns Contagious? Another fun example inspired by a Mythbusters episode.

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Reader Interactions

January 14, 2024 at 8:43 am

Hello professor Jim, how are you doing! Pls. What are the properties of a population and their examples? Thanks for your time and understanding.

January 14, 2024 at 12:57 pm

Please read my post about Populations vs. Samples for more information and examples.

Also, please note there is a search bar in the upper-right margin of my website. Use that to search for topics.

July 5, 2023 at 7:05 am

Hello, I have a question as I read your post. You say in p-values section

“P-values are the probability that you would obtain the effect observed in your sample, or larger, if the null hypothesis is correct. In simpler terms, p-values tell you how strongly your sample data contradict the null. Lower p-values represent stronger evidence against the null.”

But according to your definition of effect, the null states that an effect does not exist, correct? So what I assume you want to say is that “P-values are the probability that you would obtain the effect observed in your sample, or larger, if the null hypothesis is **incorrect**.”

July 6, 2023 at 5:18 am

Hi Shrinivas,

The correct definition of p-value is that it is a probability that exists in the context of a true null hypothesis. So, the quotation is correct in stating “if the null hypothesis is correct.”

Essentially, the p-value tells you the likelihood of your observed results (or more extreme) if the null hypothesis is true. It gives you an idea of whether your results are surprising or unusual if there is no effect.

Hence, with sufficiently low p-values, you reject the null hypothesis because it’s telling you that your sample results were unlikely to have occurred if there was no effect in the population.

I hope that helps make it more clear. If not, let me know I’ll attempt to clarify!

May 8, 2023 at 12:47 am

Thanks a lot Ny best regards

May 7, 2023 at 11:15 pm

Hi Jim Can you tell me something about size effect? Thanks

May 8, 2023 at 12:29 am

Here’s a post that I’ve written about Effect Sizes that will hopefully tell you what you need to know. Please read that. Then, if you have any more specific questions about effect sizes, please post them there. Thanks!

January 7, 2023 at 4:19 pm

Hi Jim, I have only read two pages so far but I am really amazed because in few paragraphs you made me clearly understand the concepts of months of courses I received in biostatistics! Thanks so much for this work you have done it helps a lot!

January 10, 2023 at 3:25 pm

Thanks so much!

June 17, 2021 at 1:45 pm

Can you help in the following question: Rocinante36 is priced at ₹7 lakh and has been designed to deliver a mileage of 22 km/litre and a top speed of 140 km/hr. Formulate the null and alternative hypotheses for mileage and top speed to check whether the new models are performing as per the desired design specifications.

April 19, 2021 at 1:51 pm

Its indeed great to read your work statistics.

I have a doubt regarding the one sample t-test. So as per your book on hypothesis testing with reference to page no 45, you have mentioned the difference between “the sample mean and the hypothesised mean is statistically significant”. So as per my understanding it should be quoted like “the difference between the population mean and the hypothesised mean is statistically significant”. The catch here is the hypothesised mean represents the sample mean.

Please help me understand this.

Regards Rajat

April 19, 2021 at 3:46 pm

Thanks for buying my book. I’m so glad it’s been helpful!

The test is performed on the sample but the results apply to the population. Hence, if the difference between the sample mean (observed in your study) and the hypothesized mean is statistically significant, that suggests that population does not equal the hypothesized mean.

For one sample tests, the hypothesized mean is not the sample mean. It is a mean that you want to use for the test value. It usually represents a value that is important to your research. In other words, it’s a value that you pick for some theoretical/practical reasons. You pick it because you want to determine whether the population mean is different from that particular value.

I hope that helps!

November 5, 2020 at 6:24 am

Jim, you are such a magnificent statistician/economist/econometrician/data scientist etc whatever profession. Your work inspires and simplifies the lives of so many researchers around the world. I truly admire you and your work. I will buy a copy of each book you have on statistics or econometrics. Keep doing the good work. Remain ever blessed

November 6, 2020 at 9:47 pm

Hi Renatus,

Thanks so much for you very kind comments. You made my day!! I’m so glad that my website has been helpful. And, thanks so much for supporting my books! 🙂

November 2, 2020 at 9:32 pm

Hi Jim, I hope you are aware of 2019 American Statistical Association’s official statement on Statistical Significance: https://www.tandfonline.com/doi/full/10.1080/00031305.2019.1583913 In case you do not bother reading the full article, may I quote you the core message here: “We conclude, based on our review of the articles in this special issue and the broader literature, that it is time to stop using the term “statistically significant” entirely. Nor should variants such as “significantly different,” “p < 0.05,” and “nonsignificant” survive, whether expressed in words, by asterisks in a table, or in some other way."

With best wishes,

November 3, 2020 at 2:09 am

I’m definitely aware of the debate surrounding how to use p-values most effectively. However, I need to correct you on one point. The link you provide is NOT a statement by the American Statistical Association. It is an editorial by several authors.

There is considerable debate over this issue. There are problems with p-values. However, as the authors state themselves, much of the problem is over people’s mindsets about how to use p-values and their incorrect interpretations about what statistical significance does and does not mean.

If you were to read my website more thoroughly, you’d be aware that I share many of their concerns and I address them in multiple posts. One of the authors’ key points is the need to be thoughtful and conduct thoughtful research and analysis. I emphasize this aspect in multiple posts on this topic. I’ll ask you to read the following three because they all address some of the authors’ concerns and suggestions. But you might run across others to read as well.

Five Tips for Using P-values to Avoid Being Misled How to Interpret P-values Correctly P-values and the Reproducibility of Experimental Results

September 24, 2020 at 11:52 pm

HI Jim, i just want you to know that you made explanation for Statistics so simple! I should say lesser and fewer words that reduce the complexity. All the best! 🙂

September 25, 2020 at 1:03 am

Thanks, Rene! Your kind words mean a lot to me! I’m so glad it has been helpful!

September 23, 2020 at 2:21 am

Honestly, I never understood stats during my entire M.Ed course and was another nightmare for me. But how easily you have explained each concept, I have understood stats way beyond my imagination. Thank you so much for helping ignorant research scholars like us. Looking forward to get hardcopy of your book. Kindly tell is it available through flipkart?

September 24, 2020 at 11:14 pm

I’m so happy to hear that my website has been helpful!

I checked on flipkart and it appears like my books are not available there. I’m never exactly sure where they’re available due to the vagaries of different distribution channels. They are available on Amazon in India.

Introduction to Statistics: An Intuitive Guide (Amazon IN) Hypothesis Testing: An Intuitive Guide (Amazon IN)

July 26, 2020 at 11:57 am

Dear Jim I am a teacher from India . I don’t have any background in statistics, and still I should tell that in a single read I can follow your explanations . I take my entire biostatistics class for botany graduates with your explanations. Thanks a lot. May I know how I can avail your books in India

July 28, 2020 at 12:31 am

Right now my books are only available as ebooks from my website. However, soon I’ll have some exciting news about other ways to obtain it. Stay tuned! I’ll announce it on my email list. If you’re not already on it, you can sign up using the form that is in the right margin of my website.

June 22, 2020 at 2:02 pm

Also can you please let me if this book covers topics like EDA and principal component analysis?

June 22, 2020 at 2:07 pm

This book doesn’t cover principal components analysis. Although, I wouldn’t really classify that as a hypothesis test. In the future, I might write a multivariate analysis book that would cover this and others. But, that’s well down the road.

My Introduction to Statistics covers EDA. That’s the largely graphical look at your data that you often do prior to hypothesis testing. The Introduction book perfectly leads right into the Hypothesis Testing book.

June 22, 2020 at 1:45 pm

Thanks for the detailed explanation. It does clear my doubts. I saw that your book related to hypothesis testing has the topics that I am studying currently. I am looking forward to purchasing it.

Regards, Take Care

June 19, 2020 at 1:03 pm

For this particular article I did not understand a couple of statements and it would great if you could help: 1)”If sample error causes the observed difference, the next time someone performs the same experiment the results might be different.” 2)”If the difference does not exist at the population level, you won’t obtain the benefits that you expect based on the sample statistics.”

I discovered your articles by chance and now I keep coming back to read & understand statistical concepts. These articles are very informative & easy to digest. Thanks for the simplifying things.

June 20, 2020 at 9:53 pm

I’m so happy to hear that you’ve found my website to be helpful!

To answer your questions, keep in mind that a central tenant of inferential statistics is that the random sample that a study drew was only one of an infinite number of possible it could’ve drawn. Each random sample produces different results. Most results will cluster around the population value assuming they used good methodology. However, random sampling error always exists and makes it so that population estimates from a sample almost never exactly equal the correct population value.

So, imagine that we’re studying a medication and comparing the treatment and control groups. Suppose that the medicine is truly not effect and that the population difference between the treatment and control group is zero (i.e., no difference.) Despite the true difference being zero, most sample estimates will show some degree of either a positive or negative effect thanks to random sampling error. So, just because a study has an observed difference does not mean that a difference exists at the population level. So, on to your questions:

1. If the observed difference is just random error, then it makes sense that if you collected another random sample, the difference could change. It could change from negative to positive, positive to negative, more extreme, less extreme, etc. However, if the difference exists at the population level, most random samples drawn from the population will reflect that difference. If the medicine has an effect, most random samples will reflect that fact and not bounce around on both sides of zero as much.

2. This is closely related to the previous answer. If there is no difference at the population level, but say you approve the medicine because of the observed effects in a sample. Even though your random sample showed an effect (which was really random error), that effect doesn’t exist. So, when you start using it on a larger scale, people won’t benefit from the medicine. That’s why it’s important to separate out what is easily explained by random error versus what is not easily explained by it.

I think reading my post about how hypothesis tests work will help clarify this process. Also, in about 24 hours (as I write this), I’ll be releasing my new ebook about Hypothesis Testing!

May 29, 2020 at 5:23 am

Hi Jim, I really enjoy your blog. Can you please link me on your blog where you discuss about Subgroup analysis and how it is done? I need to use non parametric and parametric statistical methods for my work and also do subgroup analysis in order to identify potential groups of patients that may benefit more from using a treatment than other groups.

May 29, 2020 at 2:12 pm

Hi, I don’t have a specific article about subgroup analysis. However, subgroup analysis is just the dividing up of a larger sample into subgroups and then analyzing those subgroups separately. You can use the various analyses I write about on the subgroups.

Alternatively, you can include the subgroups in regression analysis as an indicator variable and include that variable as a main effect and an interaction effect to see how the relationships vary by subgroup without needing to subdivide your data. I write about that approach in my article about comparing regression lines . This approach is my preferred approach when possible.

April 19, 2020 at 7:58 am

sir is confidence interval is a part of estimation?

April 17, 2020 at 3:36 pm

Sir can u plz briefly explain alternatives of hypothesis testing? I m unable to find the answer

April 18, 2020 at 1:22 am

Assuming you want to draw conclusions about populations by using samples (i.e., inferential statistics ), you can use confidence intervals and bootstrap methods as alternatives to the traditional hypothesis testing methods.

March 9, 2020 at 10:01 pm

Hi JIm, could you please help with activities that can best teach concepts of hypothesis testing through simulation, Also, do you have any question set that would enhance students intuition why learning hypothesis testing as a topic in introductory statistics. Thanks.

March 5, 2020 at 3:48 pm

Hi Jim, I’m studying multiple hypothesis testing & was wondering if you had any material that would be relevant. I’m more trying to understand how testing multiple samples simultaneously affects your results & more on the Bonferroni Correction

March 5, 2020 at 4:05 pm

I write about multiple comparisons (aka post hoc tests) in the ANOVA context . I don’t talk about Bonferroni Corrections specifically but I cover related types of corrections. I’m not sure if that exactly addresses what you want to know but is probably the closest I have already written. I hope it helps!

January 14, 2020 at 9:03 pm

Thank you! Have a great day/evening.

January 13, 2020 at 7:10 pm

Any help would be greatly appreciated. What is the difference between The Hypothesis Test and The Statistical Test of Hypothesis?

January 14, 2020 at 11:02 am

They sound like the same thing to me. Unless this is specialized terminology for a particular field or the author was intending something specific, I’d guess they’re one and the same.

April 1, 2019 at 10:00 am

so these are the only two forms of Hypothesis used in statistical testing?

April 1, 2019 at 10:02 am

Are you referring to the null and alternative hypothesis? If so, yes, that’s those are the standard hypotheses in a statistical hypothesis test.

April 1, 2019 at 9:57 am

year very insightful post, thanks for the write up

October 27, 2018 at 11:09 pm

hi there, am upcoming statistician, out of all blogs that i have read, i have found this one more useful as long as my problem is concerned. thanks so much

October 27, 2018 at 11:14 pm

Hi Stano, you’re very welcome! Thanks for your kind words. They mean a lot! I’m happy to hear that my posts were able to help you. I’m sure you will be a fantastic statistician. Best of luck with your studies!

October 26, 2018 at 11:39 am

Dear Jim, thank you very much for your explanations! I have a question. Can I use t-test to compare two samples in case each of them have right bias?

October 26, 2018 at 12:00 pm

Hi Tetyana,

You’re very welcome!

The term “right bias” is not a standard term. Do you by chance mean right skewed distributions? In other words, if you plot the distribution for each group on a histogram they have longer right tails? These are not the symmetrical bell-shape curves of the normal distribution.

If that’s the case, yes you can as long as you exceed a specific sample size within each group. I include a table that contains these sample size requirements in my post about nonparametric vs parametric analyses .

Bias in statistics refers to cases where an estimate of a value is systematically higher or lower than the true value. If this is the case, you might be able to use t-tests, but you’d need to be sure to understand the nature of the bias so you would understand what the results are really indicating.

I hope this helps!

April 2, 2018 at 7:28 am

Simple and upto the point 👍 Thank you so much.

April 2, 2018 at 11:11 am

Hi Kalpana, thanks! And I’m glad it was helpful!

March 26, 2018 at 8:41 am

Am I correct if I say: Alpha – Probability of wrongly rejection of null hypothesis P-value – Probability of wrongly acceptance of null hypothesis

March 28, 2018 at 3:14 pm

You’re correct about alpha. Alpha is the probability of rejecting the null hypothesis when the null is true.

Unfortunately, your definition of the p-value is a bit off. The p-value has a fairly convoluted definition. It is the probability of obtaining the effect observed in a sample, or more extreme, if the null hypothesis is true. The p-value does NOT indicate the probability that either the null or alternative is true or false. Although, those are very common misinterpretations. To learn more, read my post about how to interpret p-values correctly .

March 2, 2018 at 6:10 pm

I recently started reading your blog and it is very helpful to understand each concept of statistical tests in easy way with some good examples. Also, I recommend to other people go through all these blogs which you posted. Specially for those people who have not statistical background and they are facing to many problems while studying statistical analysis.

Thank you for your such good blogs.

March 3, 2018 at 10:12 pm

Hi Amit, I’m so glad that my blog posts have been helpful for you! It means a lot to me that you took the time to write such a nice comment! Also, thanks for recommending by blog to others! I try really hard to write posts about statistics that are easy to understand.

January 17, 2018 at 7:03 am

I recently started reading your blog and I find it very interesting. I am learning statistics by my own, and I generally do many google search to understand the concepts. So this blog is quite helpful for me, as it have most of the content which I am looking for.

January 17, 2018 at 3:56 pm

Hi Shashank, thank you! And, I’m very glad to hear that my blog is helpful!

January 2, 2018 at 2:28 pm

thank u very much sir.

January 2, 2018 at 2:36 pm

You’re very welcome, Hiral!

November 21, 2017 at 12:43 pm

Thank u so much sir….your posts always helps me to be a #statistician

November 21, 2017 at 2:40 pm

Hi Sachin, you’re very welcome! I’m happy that you find my posts to be helpful!

November 19, 2017 at 8:22 pm

great post as usual, but it would be nice to see an example.

November 19, 2017 at 8:27 pm

Thank you! At the end of this post, I have links to four other posts that show examples of hypothesis tests in action. You’ll find what you’re looking for in those posts!

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• Knowledge Base

Methodology

• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

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

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

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

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

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

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

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

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Step 1. Ask a question

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

Step 2. Do some preliminary research

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

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

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

4. Refine your hypothesis

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

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

5. Phrase your hypothesis in three ways

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

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

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

6. Write a null hypothesis

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

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

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

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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 Alternative 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 in 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].

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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 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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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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What kind of test to use for 2 discrete variables?

I'm trying to carry out a hypothesis test for data that has the following structure.

Total number of correct answers out of 6 questions for each subject that only has a HS diploma:

3, 2, 3, 5, ... (there are 16 total subjects)

Total number of correct answers out of 6 questions for each subject that has a College degree:

5, 5, 3, 6, ... (there are 12 total subjects)

I'm not sure if this is relevant for my question but the data was obtained by asking 28 random subjects their education level and then recording how many out of 6 questions they answered correctly.

I used a 2 sample t-test to see if the population means differ for both groups. Is there a better method for this type of data? I'm asking because the data is discrete and I'm not sure if the t-test is best suited for this kind of data.

Also the t-test would tell me if the population means differ but lets say the question of interest is whether or not education (HS vs College) is associated with literacy (total number of correct questions), what kind of test would best answer that question?

• hypothesis-testing
• statistical-significance

3 Answers 3

Right, considering the modest sample sizes and discreteness of the data, there could be some question about applying the two-sample t test here. The t test formally assumes that the data is normally distributed, which of course is not true here since the data is discrete. For large sample sizes, the t test is robust against violations of the normality assumption, but for smaller sample sizes it is less so. It may still be all right here, since the sample size is not excessively small, and especially because the data is bounded (between 0 and 6) so there can be no concerns about heavy tails or outliers (which otherwise would pose the biggest threats to the accuracy of the t test). The Wilcoxon rank sum test is an alternative which could certainly be applied, and which would avoid any concerns about distributional assumptions, although it may provide slightly less power than the t test. If you use the Wilcoxon rank sum test here, be sure to use an implementation that properly handles ties, as there are many ties here.

On the other hand, summarizing the data as counts of correct answers already discards some potentially valuable information. Particularly if the questions have varying degrees of difficulty and quality, you may be able to get more out of the data and construct a more powerful hypothesis test by applying a method based on item response theory , although this would require a more complex analysis.

Edit : As another answer mentions, the chi-square test is another option here. A drawback of the chi-square test is that it ignores the ordering of the 7 possible responses (0-6), which could lead to a less powerful test. The results of the chi-square test are also less straightforward to interpret, since if the test rejects the null hypothesis, it might be because the two populations have different means, or it could be because they have different variances, or because there is some other kind of difference between the two distributions. Another way of looking at it is that the chi-square test has a more specific null hypothesis, namely that the two distributions are identical, compared to the null hypothesis of the t test that the two distributions have the same mean. The consequence of this is that the chi-square test also has a broader alternative hypothesis, one which is less focused on detecting differences in the means, and this explains why it would be expected to be less powerful if differences in the means is what you are looking for.

Let's take a look at some simulations :) For example, I've checked a binary variable applying t-test, bootstrap, chi-square test and z-test for proportions.

I've measured power by generating multiple random samples of 1 and 0 as:

So, p1 and p2 define shares of 1-s in two samples.

Bootstrap is made like:

T-test and chi-square test have been taken from scipy.stats. Z-test has been implemented by myself.

So, I have generated a lot of samples (10 000 iterations, generating two independent samples on each) and checked in what share of iterations each test shows p-value <= 0.05.

There is something strange about chi-square test as it shows much more false positives for p1 = 0.5, p2 = 0.5 (two fair coins) case:

And the strange thing is that during the simulation series of 10000 runs for sample size = 1000 chi-square test shows a strange p-value distribution:

while it should be a near-uniform one.

Link to a notebook (sorry, it's little messy).

• $\begingroup$ Nice answer (+1). I have corrected some very trivial typos and have the further suggestion that "threatens" here perhaps means "concerns" or "worries". $\endgroup$ –  Nick Cox Commented Dec 16, 2022 at 14:44
• $\begingroup$ Yeah :) Thanks a lot! $\endgroup$ –  Vladislav Commented Dec 16, 2022 at 14:58

This seems like a typical situation for the Chi-squared test for independence. It tests whether two categorical properties are dependent or not.

In your case Property $A=\{HS, College\}$ and Property $B = \{0,\dots,6\}$.

Your null hypothesis $H_0$ would be $P(a,b) = P_A(a)P_B(b)$ (so not dependent) and your alternative $H_1$ would be $P(a,b) \neq P_A(a)P_B(b)$ (so dependent).

The idea of the test is the following. It compares the frequencies of your sample regarding property A and B with the expected ones where the two properties would be independent.

For example, you would have 5 people from HS with 4 right answers. And in total 7 people who had 4 answers right. So you would compare $$\frac{\text{amount of people with }A=HS, B=4}{\text{total amount of people}} = \frac{5}{28} \approx 0.18$$ with $$\frac{\text{amount of people with } A = HS}{\text{total amount of people}} \times \frac{\text{amount of people with } B=4}{\text{total amount of people}} = \frac{16 \times 7}{28\times28} \approx 0.15$$

It turns out there is a statistic using this frequencies which converges towards the $X^2_{(a-1)(b-1)}$ distribution (a Chi square with $(a-1)(b-1)$ degrees of freedom, where $a = |A|$ and $b = |B|$). So $a = 2$ and $b = 7$ for your scenario.

Check out this source: https://stattrek.com/chi-square-test/independence.aspx for the formulas and more explanations.

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6: Hypothesis Testing, Part 2

• Identify Type I and Type II errors
• Select an appropriate significance level (i.e., \(\alpha\) level) for a given scenario
• Explain the problems associated with conducting multiple tests
• Interpret the results of a hypothesis test in terms of practical significance
• Distinguish between practical significance and statistical significance
• Explain how changing different aspects of a research study would change the statistical power of the tests conducted
• Compare and contrast confidence intervals and hypothesis tests

Last week you learned how to conduct a hypothesis testing using randomization procedures in StatKey. This week we are going to delve a bit deeper into hypothesis testing. Concepts such as errors, significance (\(\alpha\)) levels, issues with multiple testing, practical significance, and statistical power apply to hypothesis tests for all the parameters that we have learned and will also apply to those that we will learn later in this course. 

6.1 - Type I and Type II Errors

When conducting a hypothesis test there are two possible decisions: reject the null hypothesis or fail to reject the null hypothesis. You should remember though, hypothesis testing uses data from a sample to make an inference about a population. When conducting a hypothesis test we do not know the population parameters. In most cases, we don't know if our inference is correct or incorrect.

When we reject the null hypothesis there are two possibilities. There could really be a difference in the population, in which case we made a correct decision. Or, it is possible that there is not a difference in the population (i.e., \(H_0\) is true) but our sample was different from the hypothesized value due to random sampling variation. In that case we made an error. This is known as a Type I error.

When we fail to reject the null hypothesis there are also two possibilities. If the null hypothesis is really true, and there is not a difference in the population, then we made the correct decision. If there is a difference in the population, and we failed to reject it, then we made a Type II error.

Rejecting \(H_0\) when \(H_0\) is really true, denoted by \(\alpha\) ("alpha") and commonly set at .05

     \(\alpha=P(Type\;I\;error)\)

Failing to reject \(H_0\) when \(H_0\) is really false, denoted by \(\beta\) ("beta")

     \(\beta=P(Type\;II\;error)\)

Example: Trial

A man goes to trial where he is being tried for the murder of his wife.

We can put it in a hypothesis testing framework. The hypotheses being tested are:

• \(H_0\) : Not Guilty
• \(H_a\) : Guilty

Type I error  is committed if we reject \(H_0\) when it is true. In other words, did not kill his wife but was found guilty and is punished for a crime he did not really commit.

Type II error  is committed if we fail to reject \(H_0\) when it is false. In other words, if the man did kill his wife but was found not guilty and was not punished.

Example: Culinary Arts Study

A group of culinary arts students is comparing two methods for preparing asparagus: traditional steaming and a new frying method. They want to know if patrons of their school restaurant prefer their new frying method over the traditional steaming method. A sample of patrons are given asparagus prepared using each method and asked to select their preference. A statistical analysis is performed to determine if more than 50% of participants prefer the new frying method:

• \(H_{0}: p = .50\)
• \(H_{a}: p>.50\)

Type I error  occurs if they reject the null hypothesis and conclude that their new frying method is preferred when in reality is it not. This may occur if, by random sampling error, they happen to get a sample that prefers the new frying method more than the overall population does. If this does occur, the consequence is that the students will have an incorrect belief that their new method of frying asparagus is superior to the traditional method of steaming.

Type II error  occurs if they fail to reject the null hypothesis and conclude that their new method is not superior when in reality it is. If this does occur, the consequence is that the students will have an incorrect belief that their new method is not superior to the traditional method when in reality it is.

6.2 - Significance Levels

As we saw in the examples on the previous page, the consequences of Type I and Type II errors vary depending on the situation. Researchers take into account the consequences of each when they are setting their \(\alpha\) level before data are even collected.

In many disciplines an \(\alpha\) level of 0.05 is standard, for example in the social sciences. There are some situations when a higher or lower \(\alpha\) level may be desirable. Pilot studies (smaller studies performed before a larger study) often use a higher \(\alpha\) level because their purpose is to gain information about the data that may be collected in a larger study; pilot studies are not typically used to make important decisions.

Studies in which making a Type I error would be more dangerous than making a Type II error may use smaller \(\alpha\) levels. For example, in medical research studies where making a Type I error could mean giving patients ineffective treatments, a smaller \(\alpha\) level may be set in order to reduce the likelihood of such a negative consequence. Lower \(\alpha\) levels mean that smaller p-values are needed to reject the null hypothesis; this makes it more difficult to reject the null hypothesis, but this also reduces the probability of committing a Type I error.

6.3 - Issues with Multiple Testing

If we are conducting a hypothesis test with an \(\alpha\) level of 0.05, then we are accepting a 5% chance of making a Type I error (i.e., rejecting the null hypothesis when the null hypothesis is really true). If we would conduct 100 hypothesis tests at a 0.05 \(\alpha\) level where the null hypotheses are really true, we would expect to reject the null and make a Type I error in about 5 of those tests. 

Later in this course you will learn about some statistical procedures that may be used instead of performing multiple tests. For example, to compare the means of more than two groups you can use an analysis of variance ("ANOVA"). To compare the proportions of more than two groups you can conduct a chi-square goodness-of-fit test. 

A related issue is publication bias. Research studies with statistically significant results are published much more often than studies without statistically significant results. This means that if 100 studies are performed in which there is really no difference in the population, the 5 studies that found statistically significant results may be published while the 95 studies that did not find statistically significant results will not be published. Thus, when you perform a review of published literature you will only read about the studies that found statistically significance results. You would not find the studies that did not find statistically significant results.

One quick method for correcting for multiple tests is to divide the alpha level by the number of tests being conducted. For instance, if you are comparing three groups using a series of three pairwise tests you could divided your overall alpha level ("family-wise alpha level") by three. If we were using a standard alpha level of 0.05, then our pairwise alpha level would be \(\frac{0.05}{3}=0.016667\). We would then compare each of our three p-values to 0.016667 to determine statistical significance. This is known as the  Bonferroni  method. This is one of the most conservative approaches to controlling for multiple tests (i.e., more likely to make a Type II error). Later in the course you will learn how to use the Tukey method when comparing the means of three or more groups, this approach is often preferred because it is more liberal.

6.4 - Practical Significance

In the last lesson, you learned how to identify statistically significant differences using hypothesis testing methods. If the p value is less than the \(\alpha\) level (typically 0.05), then the results are  statistically significant . Results are said to be statistically significant when the difference between the hypothesized population parameter and observed sample statistic is large enough to conclude that it is unlikely to have occurred by chance. 

Practical significance  refers to the magnitude of the difference, which is known as the  effect size . Results are practically significant when the difference is large enough to be meaningful in real life. What is meaningful may be subjective and may depend on the context.

Note that statistical significance is directly impacted by sample size. Recall that there is an inverse relationship between sample size and the standard error (i.e., standard deviation of the sampling distribution). Very small differences will be statistically significant with a very large sample size. Thus, when results are statistically significant it is important to also examine practical significance. Practical significance is not directly influenced by sample size.

Example: Weight-Loss Program

Researchers are studying a new weight-loss program. Using a large sample they construct a 95% confidence interval for the mean amount of weight loss after six months on the program to be [0.12, 0.20]. All measurements were taken in pounds. Note that this confidence interval does not contain 0, so we know that their results were statistically significant at a 0.05 alpha level. However, most people would say that the results are not practically significant because a six month weight-loss program should yield a mean weight loss much greater than the one observed in this study. 

Effect Size

For some tests there are commonly used measures of effect size. For example, when comparing the difference in two means we often compute Cohen's \(d\) which is the difference between the two observed sample means in standard deviation units:

\[d=\frac{\overline x_1 - \overline x_2}{s_p}\]

Where \(s_p\) is the pooled standard deviation

\[s_p= \sqrt{\frac{(n_1-1)s_1^2 + (n_2 -1)s_2^2}{n_1+n_2-2}}\]

Below are commonly used standards when interpreting Cohen's \(d\):

For a single mean, you can compute the difference between the observed mean and hypothesized mean in standard deviation units: \[d=\frac{\overline x - \mu_0}{s}\]

For correlation and regression we can compute \(r^2\) which is known as the coefficient of determination. This is the proportion of shared variation. We will learn more about \(r^2\) when we study simple linear regression and correlation at the end of this course.

Example: SAT-Math Scores

Research question :  Are SAT-Math scores at one college greater than the known population mean of 500?

\(H_0\colon \mu = 500\)

\(H_a\colon \mu >500\)

Data are collected from a random sample of 1,200 students at that college. In that sample, \(\overline{x}=506\) and the sample standard deviation was 100. A one-sample mean test was performed and the resulting p-value was 0.0188. Because \(p \leq \alpha\), the null hypothesis should be rejected. These results are statistically significant. There is evidence that the population mean is greater than 500.

But, let's also consider practical significance. The difference between an SAT-Math score 500 and an SAT-Math score of 506 is very small. With a standard deviation of 100, this difference is only \(\frac{506-500}{100}=0.06\) standard deviations. In most cases, this would not be considered practically significant. 

Example: Commute Times

Research question:  Are the mean commute times different in Atlanta and St. Louis?

Using the dataset built in to StatKey , a two-tailed randomization test was conducted resulting in a p value < 0.001. Because the null hypothesis was rejected, the results are said to be statistically significant.

Practical significance can be examined by computing Cohen's d. We'll use the equations from above:

First, we compute the pooled standard deviation:

\[s_p= \sqrt{\frac{(500-1)20.718^2 + (500-1)14.232^2}{500+500-2}}\]

\[s_p= \sqrt{\frac{(499)(429.236)+ (499)(202.550)}{998}}\]

\[s_p= \sqrt{\frac{214188.527+ 101072.362}{998}}\]

\[s_p= \sqrt{\frac{315260.853}{998}}\]

\[s_p= \sqrt{315.893}\]

\[s_p= 17.773\]

Note: The pooled standard deviation should always be between the two sample standard deviations.

Next, we can compute Cohen's d:

\[d=\frac{29.110-21.970}{17.773}\]

\[d=\frac{7.14}{17.773}\]

\[d= 0.402\]

The mean commute time in Atlanta was 0.402 standard deviations greater than the mean commute time in St. Louis. Using the guidelines for interpreting Cohen's d in the table above, this is a small effect size. 

6.5 - Power

The probability of rejecting the null hypothesis, given that the null hypothesis is false, is known as power. In other words, power is the probability of correctly rejecting \(H_0\).

The power of a test can be increased in a number of ways, for example increasing the sample size, decreasing the standard error, increasing the difference between the sample statistic and the hypothesized parameter, or increasing the alpha level. Using a directional test (i.e., left- or right-tailed) as opposed to a two-tailed test would also increase power. 

When we increase the sample size, decrease the standard error, or increase the difference between the sample statistic and hypothesized parameter, the p value decreases, thus making it more likely that we reject the null hypothesis. When we increase the alpha level, there is a larger range of p values for which we would reject the null hypothesis. Going from a two-tailed to a one-tailed test cuts the p value in half. In all of these cases, we say that statistically power is increased. 

There is a relationship between \(\alpha\) and \(\beta\). If the sample size is fixed, then decreasing \(\alpha\) will increase \(\beta\). If we want both \(\alpha\) and \(\beta\) to decrease (i.e., decreasing the likelihood of both Type I and Type II errors), then we should increase the sample size.

The probability of committing a Type II error is known as \(\beta\).

\(Power+\beta=1\)

\(Power=1-\beta\)

If power increases then \(\beta\) must decrease. So, if the power of a statistical test is increased, for example by increasing the sample size, the probability of committing a Type II error decreases.

No. When we perform a hypothesis test, we only set the Type I error rate (i.e., alpha level) and guard against it. Thus, we can only present the strength of evidence against the null hypothesis. We can sidestep the concern about Type II error if the conclusion never mentions that the null hypothesis is accepted. When the null hypothesis cannot be rejected, there are two possible cases:

1) The null hypothesis is really true.

2) The sample size is not large enough to reject the null hypothesis (i.e., statistical power is too low).

The result of the study was to fail to reject the null hypothesis. In reality, the null hypothesis was false. This is a Type II error.

6.6 - Confidence Intervals & Hypothesis Testing

Confidence intervals and hypothesis tests are similar in that they are both inferential methods that rely on an approximated sampling distribution. Confidence intervals use data from a sample to estimate a population parameter. Hypothesis tests use data from a sample to test a specified hypothesis. Hypothesis testing requires that we have a hypothesized parameter. 

The simulation methods used to construct bootstrap distributions and randomization distributions are similar. One primary difference is a bootstrap distribution is centered on the observed sample statistic while a randomization distribution is centered on the value in the null hypothesis. 

In Lesson 4, we learned confidence intervals contain a range of reasonable estimates of the population parameter. All of the confidence intervals we constructed in this course were two-tailed. These two-tailed confidence intervals go hand-in-hand with the two-tailed hypothesis tests we learned in Lesson 5. The conclusion drawn from a two-tailed confidence interval is usually the same as the conclusion drawn from a two-tailed hypothesis test. In other words, if the the 95% confidence interval contains the hypothesized parameter, then a hypothesis test at the 0.05 \(\alpha\) level will almost always fail to reject the null hypothesis. If the 95% confidence interval does not contain the hypothesize parameter, then a hypothesis test at the 0.05 \(\alpha\) level will almost always reject the null hypothesis.

Example: Mean

This example uses the Body Temperature dataset built in to StatKey for constructing a  bootstrap confidence interval and conducting a randomization test . 

Let's start by constructing a 95% confidence interval using the percentile method in StatKey:

The 95% confidence interval for the mean body temperature in the population is [98.044, 98.474].

Now, what if we want to know if there is enough evidence that the mean body temperature is different from 98.6 degrees? We can conduct a hypothesis test. Because 98.6 is not contained within the 95% confidence interval, it is not a reasonable estimate of the population mean. We should expect to have a p value less than 0.05 and to reject the null hypothesis.

\(H_0: \mu=98.6\)

\(H_a: \mu \ne 98.6\)

\(p = 2*0.00080=0.00160\)

\(p \leq 0.05\), reject the null hypothesis

There is evidence that the population mean is different from 98.6 degrees. 

Selecting the Appropriate Procedure

The decision of whether to use a confidence interval or a hypothesis test depends on the research question. If we want to estimate a population parameter, we use a confidence interval. If we are given a specific population parameter (i.e., hypothesized value), and want to determine the likelihood that a population with that parameter would produce a sample as different as our sample, we use a hypothesis test. Below are a few examples of selecting the appropriate procedure. 

Example: Cheese Consumption

Research question: How much cheese (in pounds) does an average American adult consume annually? 

What is the appropriate inferential procedure? 

Cheese consumption, in pounds, is a quantitative variable. We have one group: American adults. We are not given a specific value to test, so the appropriate procedure here is a  confidence interval for a single mean .

Example: Age

Research question:  Is the average age in the population of all STAT 200 students greater than 30 years?

There is one group: STAT 200 students. The variable of interest is age in years, which is quantitative. The research question includes a specific population parameter to test: 30 years. The appropriate procedure is a  hypothesis test for a single mean .

For each research question, identify the variables, the parameter of interest and decide on the the appropriate inferential procedure.

Research question:  How strong is the correlation between height (in inches) and weight (in pounds) in American teenagers?

There are two variables of interest: (1) height in inches and (2) weight in pounds. Both are quantitative variables. The parameter of interest is the correlation between these two variables.

We are not given a specific correlation to test. We are being asked to estimate the strength of the correlation. The appropriate procedure here is a  confidence interval for a correlation . 

Research question:  Are the majority of registered voters planning to vote in the next presidential election?

The parameter that is being tested here is a single proportion. We have one group: registered voters. "The majority" would be more than 50%, or p>0.50. This is a specific parameter that we are testing. The appropriate procedure here is a  hypothesis test for a single proportion .

Research question:  On average, are STAT 200 students younger than STAT 500 students?

We have two independent groups: STAT 200 students and STAT 500 students. We are comparing them in terms of average (i.e., mean) age.

If STAT 200 students are younger than STAT 500 students, that translates to \(\mu_{200}<\mu_{500}\) which is an alternative hypothesis. This could also be written as \(\mu_{200}-\mu_{500}<0\), where 0 is a specific population parameter that we are testing. 

The appropriate procedure here is a  hypothesis test for the difference in two means .

Research question:  On average, how much taller are adult male giraffes compared to adult female giraffes?

There are two groups: males and females. The response variable is height, which is quantitative. We are not given a specific parameter to test, instead we are asked to estimate "how much" taller males are than females. The appropriate procedure is a  confidence interval for the difference in two means .

Research question:  Are STAT 500 students more likely than STAT 200 students to be employed full-time?

There are two independent groups: STAT 500 students and STAT 200 students. The response variable is full-time employment status which is categorical with two levels: yes/no.

If STAT 500 students are more likely than STAT 200 students to be employed full-time, that translates to \(p_{500}>p_{200}\) which is an alternative hypothesis. This could also be written as \(p_{500}-p_{200}>0\), where 0 is a specific parameter that we are testing. The appropriate procedure is a  hypothesis test for the difference in two proportions.

Research question:  Is there is a relationship between outdoor temperature (in Fahrenheit) and coffee sales (in cups per day)?

There are two variables here: (1) temperature in Fahrenheit and (2) cups of coffee sold in a day. Both variables are quantitative. The parameter of interest is the correlation between these two variables.

If there is a relationship between the variables, that means that the correlation is different from zero. This is a specific parameter that we are testing. The appropriate procedure is a  hypothesis test for a correlation . 

6.7 - Lesson 6 Summary

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• Understanding How to Approach Statistics Homework with SPSS

Understanding How to Approach Complex Statistics Homework Using SPSS

In your journey as a student, you will likely encounter statistics homework that require you to perform complex analyses using tools like SPSS. Whether you’re new to the software or have some experience, mastering the skills to solve these types of assignments is crucial. This blog will guide you on how to solve your SPSS homework effectively, breaking down the process into manageable steps, and providing tips that apply to similar problems.

1. Understanding the Problem Statement

Before diving into any statistical analysis, it’s essential to understand the problem statement clearly. Take the time to read and interpret the requirements of your assignment. Typically, assignments involve analyzing relationships between variables, conducting hypothesis testing , checking for assumptions (like normality and homoscedasticity), and interpreting the results.

For instance, in a typical assignment, you might be asked to:

• Regress one variable on another and interpret the results.
• Test for assumptions like heteroscedasticity and normality.
• Determine if the model is well-specified.
• Analyze the impact of additional variables on the results.

2. Organizing and Preparing Your Data

The first step in any analysis is to ensure your data is well-organized. You might be given a dataset in SPSS format or need to import it from Excel or another source. Make sure the variables are labeled correctly and that you understand what each represents.

• Clean the data: Remove any outliers, handle missing data, and ensure your dataset is ready for analysis.
• Understand your variables: Identify dependent and independent variables. For example, if you’re analyzing the relationship between IQ and wages, IQ would be your independent variable, and wage would be your dependent variable.

3. Performing Regressions and Analyzing Relationships

Regression analysis is at the heart of many statistical assignments. You may be required to perform simple linear regression, multiple regression, or even log transformations of variables.

• Simple Linear Regression: This is when you regress one variable on another to determine their relationship. For example, regressing wage on IQ might reveal that a 1-point increase in IQ results in an $8.30 increase in weekly wage.
• Multiple Regression: When more than one independent variable is involved, you’ll use multiple regression. For instance, you might regress exam scores on both class hours and homework submission rates to see how each factor contributes to exam performance.
• Log Transformations: Sometimes, transforming your variables (e.g., taking the logarithm) can help stabilize variance and improve model fit.

4. Hypothesis Testing: Understanding Significance

A key part of regression analysis involves testing hypotheses. You’ll often be asked to test whether the relationship between variables is statistically significant. This involves setting up null (H0) and alternative (H1) hypotheses, performing the regression, and interpreting the p-values.

• Interpreting Results: A p-value less than 0.05 typically indicates that the independent variable has a significant impact on the dependent variable, allowing you to reject the null hypothesis.
• R-Squared Value: This value tells you how much of the variation in the dependent variable is explained by the independent variable(s). For example, an R-squared value of 0.095 means that 9.5% of the variation in wage is explained by IQ.

5. Checking for Assumptions: Heteroscedasticity, Normality, and Linearity

Your analysis isn’t complete until you check whether the assumptions underlying the regression are met. Common assumptions include:

• Heteroscedasticity: This occurs when the variance of the residuals (errors) is not constant across levels of the independent variable. A Breusch-Pagan test can help determine if heteroscedasticity is present.
• Normality of Residuals: Use a Normal Q-Q plot and histograms to check if the residuals are normally distributed. If they are not, this could indicate problems with your model.
• Linearity: Scatter plots of residuals versus predicted values can reveal if the relationship between variables is linear. A non-linear relationship suggests your model might be misspecified.

6. Model Specification: Is Your Model Correct?

Model specification involves ensuring that your chosen model is appropriate for the data. A misspecified model might include the wrong variables or exclude important ones, leading to biased estimates.

• Adding or Removing Variables: You might be asked to test if adding another variable improves the model. For instance, including homework submission rate along with class hours when predicting exam scores could provide a more comprehensive understanding of the factors that influence performance.

7. Presenting and Interpreting Your Findings

Once your analysis is complete, you’ll need to present your findings clearly and concisely. This includes reporting:

• The estimated coefficients and their interpretation (e.g., a coefficient of 8.298 for IQ suggests that each additional IQ point increases wage by $8.30).
• The statistical significance of your variables (e.g., “The p-value of 0.000 indicates that IQ significantly affects wage at a 5% significance level”).
• Any issues with assumptions (e.g., “The model exhibits heteroscedasticity based on the Breusch-Pagan test”).

8. Practical Tips for Solving Statistical Assignments

Here are some additional tips to help you navigate similar assignments:

• Use SPSS Effectively: Learn the basics of SPSS, such as importing data, running regressions, and interpreting output. The software has built-in functions for most statistical tests, making your work easier.
• Understand the Theory: Statistical analysis isn’t just about plugging numbers into software. Make sure you understand the underlying theory behind concepts like regression, hypothesis testing, and model assumptions.
• Double-Check Your Work: Always verify your results by cross-referencing them with manual calculations or by running similar tests using different methods.
• Practice, Practice, Practice: The more you practice working on different datasets and assignments, the more confident you’ll become. Work through sample problems, either from your coursework or online, to build your skills.

9. Common Challenges and How to Overcome Them

Statistics assignments can be challenging, especially when dealing with complex datasets and models. Here are some common challenges students face:

• Interpreting SPSS Output: The output generated by SPSS can be overwhelming. Focus on the key results, such as coefficients, p-values, R-squared values, and test statistics.
• Multicollinearity: This occurs when independent variables are highly correlated with each other, leading to unreliable estimates. If you encounter high VIFs (Variance Inflation Factors), consider removing one of the correlated variables.
• Non-Normal Residuals: If your residuals are not normally distributed, consider transforming your variables or using robust regression techniques.

10. Expanding Your Analysis: Adding More Variables and Testing Interactions

In more advanced assignments, you might be asked to explore interactions between variables or include additional factors in your regression model. For example, when analyzing exam scores, you could include variables like hours of revision, sleep, and extracurricular activities.

• Interaction Terms: Interaction terms allow you to examine whether the effect of one variable depends on another. For instance, you might explore whether the impact of class hours on exam scores changes depending on how much homework a student completes.
• Dummy Variables: If your assignment involves categorical variables, you’ll need to include dummy variables in your regression model. For example, in analyzing wage data, you might include a dummy variable for gender (male = 1, female = 0).

11. Final Thoughts: Continuous Learning and Improvement

Statistics is a field that rewards continuous learning. As you progress in your studies, you’ll encounter more complex models, data analysis techniques, and software tools. Keep challenging yourself to go beyond the basics and explore advanced topics like time-series analysis, logistic regression, and panel data models.

Remember, each assignment you complete is an opportunity to improve your skills. By following a systematic approach—understanding the problem, organizing your data, performing analyses, checking assumptions, and interpreting results—you’ll be well-prepared to tackle any statistical assignment that comes your way.

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Understanding how to approach statistics homework with spss submit your homework, attached files.

A dataset of cognitive tests with various variables for dyslexic children, Iran 2023

Description.

The primary hypothesis posited that the Program for Attentive Rehabilitation of Inhibition and Selective Attention (PARISA) would significantly improve inhibitory control and reading abilities in students with dyslexia. This study's dataset shows all the test results of various cognitive tests measuring inhibitory control, cognitive flexibility, and reading ability. This data shows that PARISA significantly improves aspects of inhibition control, cognitive flexibility, and reading ability in students with dyslexia. The Excel file of the dataset neatly presents all of the variables and related data for the active control group and the experiment group in the pretest and posttest stages. There are also several separate notepads with raw data for each test and participant.  This study employed a semi-experimental design, specifically a pre-test-post-test format with an active control group. The research population comprised Iranian elementary school students aged 8 to 11 years old with dyslexia, sampled in 2023. Using purposive sampling, 30 students (16 girls and 14 boys) from the Tehran metropolitan area were selected. These participants were randomly assigned to either the intervention or active control group. Participants were excluded if they had any documented history of psychiatric, neurological, or developmental disorders, as reported by the admitting institution. The assessment tools utilized in both the pre-test and post-test phases included the Farsi Apra reading test, WCST, the go/no-go test, and the Stroop task. Any willing researcher can interpret this dataset by entering pretest and posttest data for each variable in a data analysis tool. The data is stored in a concise and clear manner for any user. Further, you can check the raw data from notepads and compare it manually with the data stored in the Excel file.  For the reading ability test, since the test materials were paper-based and in Farsi, therefore, no analysis can be derived from the data. However, I have put all the data required in the Excel file.

Steps to reproduce

This study employed a semi-experimental design, specifically a pre-test-post-test format with an active control group. The research population comprised Iranian elementary school students aged 8 to 11 years old with dyslexia, sampled in 2023. Using purposive sampling, 30 students (16 girls and 14 boys) from Tehran metropolitan area were selected. These participants were randomly assigned to either the intervention or active control group. Participants were excluded if they had any documented history of psychiatric, neurological, or developmental disorders, as reported by the admitting institution. The assessment tools utilized in both the pre-test and post-test phases included the Farsi Apra reading test, WCST, the go/no-go test, and the Stroop task. Attentive Rehabilitation of Recognition of Emotional Face (AREF) was implemented on active control group to preserve their inhibition control abilities. The inhibitory control training protocol, known as the Program for Attentive Rehabilitation of Inhibition and Selective Attention (PARISA), was implemented on the experimental group. Apra Farsi reading ability assessment test In this test, the materials are extracted from the list of words found in the Persian books of the first to fifth grade. In this test, participants read a series of cards with Farsi texts. The quality of their reading, pronunciation, speed, accuracy, and the number of mistakes is evaluated. The go/no-go task The go/no-go task has been mostly used to assess the response inhibition. In other words, the inability to inhibit the response to a ‘no-go’ stimulus is regarded as inhibition failure, which is caused by a deficit of attention or an inability in the speed-accuracy interaction. The Stroop task The Stroop task, or color-word test, is a task that measures interference inhibition. Wisconsin Card Sorting Test Wisconsin Card Sorting Test (WCST) was devised in 1948 by Grant and Berg as an index of abstract reasoning, concept formation, and response strategies to changing contextual contingencies. This assessment is designed to evaluate cognitive flexibility. Statistical analysis The normality of the data was tested using Kolmogorov-Smirnov (P > 0.05). The normality of error variances was justified by Levene’s test (P > 0.05). The assumption of homogeneity of covariance matrices determined that covariance matrices are equal and the ANCOVA can be calculated. For data analysis, SPSS-27 was used.

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