greater than (>) less than (<)
H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.
H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ 30
H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30
A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.
H 0 : The drug reduces cholesterol by 25%. p = 0.25
H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25
We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:
H 0 : μ = 2.0
H a : μ ≠ 2.0
We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 66 H a : μ __ 66
We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:
H 0 : μ ≥ 5
H a : μ < 5
We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 45 H a : μ __ 45
In an issue of U.S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.
H 0 : p ≤ 0.066
H a : p > 0.066
On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : p __ 0.40 H a : p __ 0.40
In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis , typically denoted with H 0 . The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.
H 0 and H a are contradictory.
Statistics By Jim
Making statistics intuitive
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 .
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 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 .
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
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
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
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.
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
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:
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!
Hypothesis testing, the null and alternative hypothesis.
In order to undertake hypothesis testing you need to express your research hypothesis as a null and alternative hypothesis. The null hypothesis and alternative hypothesis are statements regarding the differences or effects that occur in the population. You will use your sample to test which statement (i.e., the null hypothesis or alternative hypothesis) is most likely (although technically, you test the evidence against the null hypothesis). So, with respect to our teaching example, the null and alternative hypothesis will reflect statements about all statistics students on graduate management courses.
The null hypothesis is essentially the "devil's advocate" position. That is, it assumes that whatever you are trying to prove did not happen ( hint: it usually states that something equals zero). For example, the two different teaching methods did not result in different exam performances (i.e., zero difference). Another example might be that there is no relationship between anxiety and athletic performance (i.e., the slope is zero). The alternative hypothesis states the opposite and is usually the hypothesis you are trying to prove (e.g., the two different teaching methods did result in different exam performances). Initially, you can state these hypotheses in more general terms (e.g., using terms like "effect", "relationship", etc.), as shown below for the teaching methods example:
Null Hypotheses (H ): | Undertaking seminar classes has no effect on students' performance. |
Alternative Hypothesis (H ): | Undertaking seminar class has a positive effect on students' performance. |
Depending on how you want to "summarize" the exam performances will determine how you might want to write a more specific null and alternative hypothesis. For example, you could compare the mean exam performance of each group (i.e., the "seminar" group and the "lectures-only" group). This is what we will demonstrate here, but other options include comparing the distributions , medians , amongst other things. As such, we can state:
Null Hypotheses (H ): | The mean exam mark for the "seminar" and "lecture-only" teaching methods is the same in the population. |
Alternative Hypothesis (H ): | The mean exam mark for the "seminar" and "lecture-only" teaching methods is not the same in the population. |
Now that you have identified the null and alternative hypotheses, you need to find evidence and develop a strategy for declaring your "support" for either the null or alternative hypothesis. We can do this using some statistical theory and some arbitrary cut-off points. Both these issues are dealt with next.
The level of statistical significance is often expressed as the so-called p -value . Depending on the statistical test you have chosen, you will calculate a probability (i.e., the p -value) of observing your sample results (or more extreme) given that the null hypothesis is true . Another way of phrasing this is to consider the probability that a difference in a mean score (or other statistic) could have arisen based on the assumption that there really is no difference. Let us consider this statement with respect to our example where we are interested in the difference in mean exam performance between two different teaching methods. If there really is no difference between the two teaching methods in the population (i.e., given that the null hypothesis is true), how likely would it be to see a difference in the mean exam performance between the two teaching methods as large as (or larger than) that which has been observed in your sample?
So, you might get a p -value such as 0.03 (i.e., p = .03). This means that there is a 3% chance of finding a difference as large as (or larger than) the one in your study given that the null hypothesis is true. However, you want to know whether this is "statistically significant". Typically, if there was a 5% or less chance (5 times in 100 or less) that the difference in the mean exam performance between the two teaching methods (or whatever statistic you are using) is as different as observed given the null hypothesis is true, you would reject the null hypothesis and accept the alternative hypothesis. Alternately, if the chance was greater than 5% (5 times in 100 or more), you would fail to reject the null hypothesis and would not accept the alternative hypothesis. As such, in this example where p = .03, we would reject the null hypothesis and accept the alternative hypothesis. We reject it because at a significance level of 0.03 (i.e., less than a 5% chance), the result we obtained could happen too frequently for us to be confident that it was the two teaching methods that had an effect on exam performance.
Whilst there is relatively little justification why a significance level of 0.05 is used rather than 0.01 or 0.10, for example, it is widely used in academic research. However, if you want to be particularly confident in your results, you can set a more stringent level of 0.01 (a 1% chance or less; 1 in 100 chance or less).
When considering whether we reject the null hypothesis and accept the alternative hypothesis, we need to consider the direction of the alternative hypothesis statement. For example, the alternative hypothesis that was stated earlier is:
Alternative Hypothesis (H ): | Undertaking seminar classes has a positive effect on students' performance. |
The alternative hypothesis tells us two things. First, what predictions did we make about the effect of the independent variable(s) on the dependent variable(s)? Second, what was the predicted direction of this effect? Let's use our example to highlight these two points.
Sarah predicted that her teaching method (independent variable: teaching method), whereby she not only required her students to attend lectures, but also seminars, would have a positive effect (that is, increased) students' performance (dependent variable: exam marks). If an alternative hypothesis has a direction (and this is how you want to test it), the hypothesis is one-tailed. That is, it predicts direction of the effect. If the alternative hypothesis has stated that the effect was expected to be negative, this is also a one-tailed hypothesis.
Alternatively, a two-tailed prediction means that we do not make a choice over the direction that the effect of the experiment takes. Rather, it simply implies that the effect could be negative or positive. If Sarah had made a two-tailed prediction, the alternative hypothesis might have been:
Alternative Hypothesis (H ): | Undertaking seminar classes has an effect on students' performance. |
In other words, we simply take out the word "positive", which implies the direction of our effect. In our example, making a two-tailed prediction may seem strange. After all, it would be logical to expect that "extra" tuition (going to seminar classes as well as lectures) would either have a positive effect on students' performance or no effect at all, but certainly not a negative effect. However, this is just our opinion (and hope) and certainly does not mean that we will get the effect we expect. Generally speaking, making a one-tail prediction (i.e., and testing for it this way) is frowned upon as it usually reflects the hope of a researcher rather than any certainty that it will happen. Notable exceptions to this rule are when there is only one possible way in which a change could occur. This can happen, for example, when biological activity/presence in measured. That is, a protein might be "dormant" and the stimulus you are using can only possibly "wake it up" (i.e., it cannot possibly reduce the activity of a "dormant" protein). In addition, for some statistical tests, one-tailed tests are not possible.
Let's return finally to the question of whether we reject or fail to reject the null hypothesis.
If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above the cut-off value, we fail to reject the null hypothesis and cannot accept the alternative hypothesis. You should note that you cannot accept the null hypothesis, but only find evidence against it.
What is the null hypothesis, how to state the null hypothesis, null hypothesis overview.
The word “null” in this context means that it’s a commonly accepted fact that researchers work to nullify . It doesn’t mean that the statement is null (i.e. amounts to nothing) itself! (Perhaps the term should be called the “nullifiable hypothesis” as that might cause less confusion).
The short answer is, as a scientist, you are required to ; It’s part of the scientific process. Science uses a battery of processes to prove or disprove theories, making sure than any new hypothesis has no flaws. Including both a null and an alternate hypothesis is one safeguard to ensure your research isn’t flawed. Not including the null hypothesis in your research is considered very bad practice by the scientific community. If you set out to prove an alternate hypothesis without considering it, you are likely setting yourself up for failure. At a minimum, your experiment will likely not be taken seriously.
Several scientists, including Copernicus , set out to disprove the null hypothesis. This eventually led to the rejection of the null and the acceptance of the alternate. Most people accepted it — the ones that didn’t created the Flat Earth Society !. What would have happened if Copernicus had not disproved the it and merely proved the alternate? No one would have listened to him. In order to change people’s thinking, he first had to prove that their thinking was wrong .
You’ll be asked to convert a word problem into a hypothesis statement in statistics that will include a null hypothesis and an alternate hypothesis . Breaking your problem into a few small steps makes these problems much easier to handle.
Step 2: Convert the hypothesis to math . Remember that the average is sometimes written as μ.
H 1 : μ > 8.2
Broken down into (somewhat) English, that’s H 1 (The hypothesis): μ (the average) > (is greater than) 8.2
Step 3: State what will happen if the hypothesis doesn’t come true. If the recovery time isn’t greater than 8.2 weeks, there are only two possibilities, that the recovery time is equal to 8.2 weeks or less than 8.2 weeks.
H 0 : μ ≤ 8.2
Broken down again into English, that’s H 0 (The null hypothesis): μ (the average) ≤ (is less than or equal to) 8.2
But what if the researcher doesn’t have any idea what will happen.
Example Problem: A researcher is studying the effects of radical exercise program on knee surgery patients. There is a good chance the therapy will improve recovery time, but there’s also the possibility it will make it worse. Average recovery times for knee surgery patients is 8.2 weeks.
Step 1: State what will happen if the experiment doesn’t make any difference. That’s the null hypothesis–that nothing will happen. In this experiment, if nothing happens, then the recovery time will stay at 8.2 weeks.
H 0 : μ = 8.2
Broken down into English, that’s H 0 (The null hypothesis): μ (the average) = (is equal to) 8.2
Step 2: Figure out the alternate hypothesis . The alternate hypothesis is the opposite of the null hypothesis. In other words, what happens if our experiment makes a difference?
H 1 : μ ≠ 8.2
In English again, that’s H 1 (The alternate hypothesis): μ (the average) ≠ (is not equal to) 8.2
That’s How to State the Null Hypothesis!
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Gonick, L. (1993). The Cartoon Guide to Statistics . HarperPerennial. Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences , Wiley.
by Marco Taboga , PhD
In a test of hypothesis , a sample of data is used to decide whether to reject or not to reject a hypothesis about the probability distribution from which the sample was extracted.
The hypothesis is called the null hypothesis, or simply "the null".
Table of contents
How is the null hypothesis tested, example 1 - proportion of defective items, measurement, test statistic, critical region, interpretation, example 2 - reliability of a production plant, rejection and failure to reject, not rejecting and accepting are not the same thing, failure to reject can be due to lack of power, rejections are easier to interpret, but be careful, takeaways - how to (and not to) formulate a null hypothesis, more examples, more details, best practices in science, keep reading the glossary.
Formulating null hypotheses and subjecting them to statistical testing is one of the workhorses of the scientific method.
Scientists in all fields make conjectures about the phenomena they study, translate them into null hypotheses and gather data to test them.
This process resembles a trial:
the defendant (the null hypothesis) is accused of being guilty (wrong);
evidence (data) is gathered in order to prove the defendant guilty (reject the null);
if there is evidence beyond any reasonable doubt, the defendant is found guilty (the null is rejected);
otherwise, the defendant is found not guilty (the null is not rejected).
Keep this analogy in mind because it helps to better understand statistical tests, their limitations, use and misuse, and frequent misinterpretation.
Before collecting the data:
we decide how to summarize the relevant characteristics of the sample data in a single number, the so-called test statistic ;
we derive the probability distribution of the test statistic under the hypothesis that the null is true (the data is regarded as random; therefore, the test statistic is a random variable);
we decide what probability of incorrectly rejecting the null we are willing to tolerate (the level of significance , or size of the test ); the level of significance is typically a small number, such as 5% or 1%.
we choose one or more intervals of values (collectively called rejection region) such that the probability that the test statistic falls within these intervals is equal to the desired level of significance; the rejection region is often a tail of the distribution of the test statistic (one-tailed test) or the union of the left and right tails (two-tailed test).
Then, the data is collected and used to compute the value of the test statistic.
A decision is taken as follows:
if the test statistic falls within the rejection region, then the null hypothesis is rejected;
otherwise, it is not rejected.
We now make two examples of practical problems that lead to formulate and test a null hypothesis.
A new method is proposed to produce light bulbs.
The proponents claim that it produces less defective bulbs than the method currently in use.
To check the claim, we can set up a statistical test as follows.
We keep the light bulbs on for 10 consecutive days, and then we record whether they are still working at the end of the test period.
The probability that a light bulb produced with the new method is still working at the end of the test period is the same as that of a light bulb produced with the old method.
100 light bulbs are tested:
50 of them are produced with the new method (group A)
the remaining 50 are produced with the old method (group B).
The final data comprises 100 observations of:
an indicator variable which is equal to 1 if the light bulb is still working at the end of the test period and 0 otherwise;
a categorical variable that records the group (A or B) to which each light bulb belongs.
We use the data to compute the proportions of working light bulbs in groups A and B.
The proportions are estimates of the probabilities of not being defective, which are equal for the two groups under the null hypothesis.
We then compute a z-statistic (see here for details) by:
taking the difference between the proportion in group A and the proportion in group B;
standardizing the difference:
we subtract the expected value (which is zero under the null hypothesis);
we divide by the standard deviation (it can be derived analytically).
The distribution of the z-statistic can be approximated by a standard normal distribution .
We decide that the level of confidence must be 5%. In other words, we are going to tolerate a 5% probability of incorrectly rejecting the null hypothesis.
The critical region is the right 5%-tail of the normal distribution, that is, the set of all values greater than 1.645 (see the glossary entry on critical values if you are wondering how this value was obtained).
If the test statistic is greater than 1.645, then the null hypothesis is rejected; otherwise, it is not rejected.
A rejection is interpreted as significant evidence that the new production method produces less defective items; failure to reject is interpreted as insufficient evidence that the new method is better.
A production plant incurs high costs when production needs to be halted because some machinery fails.
The plant manager has decided that he is not willing to tolerate more than one halt per year on average.
If the expected number of halts per year is greater than 1, he will make new investments in order to improve the reliability of the plant.
A statistical test is set up as follows.
The reliability of the plant is measured by the number of halts.
The number of halts in a year is assumed to have a Poisson distribution with expected value equal to 1 (using the Poisson distribution is common in reliability testing).
The manager cannot wait more than one year before taking a decision.
There will be a single datum at his disposal: the number of halts observed during one year.
The number of halts is used as a test statistic. By assumption, it has a Poisson distribution under the null hypothesis.
The manager decides that the probability of incorrectly rejecting the null can be at most 10%.
A Poisson random variable with expected value equal to 1 takes values:
larger than 1 with probability 26.42%;
larger than 2 with probability 8.03%.
Therefore, it is decided that the critical region will be the set of all values greater than or equal to 3.
If the test statistic is strictly greater than or equal to 3, then the null is rejected; otherwise, it is not rejected.
A rejection is interpreted as significant evidence that the production plant is not reliable enough (the average number of halts per year is significantly larger than tolerated).
Failure to reject is interpreted as insufficient evidence that the plant is unreliable.
This section discusses the main problems that arise in the interpretation of the outcome of a statistical test (reject / not reject).
When the test statistic does not fall within the critical region, then we do not reject the null hypothesis.
Does this mean that we accept the null? Not really.
In general, failure to reject does not constitute, per se, strong evidence that the null hypothesis is true .
Remember the analogy between hypothesis testing and a criminal trial. In a trial, when the defendant is declared not guilty, this does not mean that the defendant is innocent. It only means that there was not enough evidence (not beyond any reasonable doubt) against the defendant.
In turn, lack of evidence can be due:
either to the fact that the defendant is innocent ;
or to the fact that the prosecution has not been able to provide enough evidence against the defendant, even if the latter is guilty .
This is the very reason why courts do not declare defendants innocent, but they use the locution "not guilty".
In a similar fashion, statisticians do not say that the null hypothesis has been accepted, but they say that it has not been rejected.
To better understand why failure to reject does not in general constitute strong evidence that the null hypothesis is true, we need to use the concept of statistical power .
The power of a test is the probability (calculated ex-ante, i.e., before observing the data) that the null will be rejected when another hypothesis (called the alternative hypothesis ) is true.
Let's consider the first of the two examples above (the production of light bulbs).
In that example, the null hypothesis is: the probability that a light bulb is defective does not decrease after introducing a new production method.
Let's make the alternative hypothesis that the probability of being defective is 1% smaller after changing the production process (assume that a 1% decrease is considered a meaningful improvement by engineers).
How much is the ex-ante probability of rejecting the null if the alternative hypothesis is true?
If this probability (the power of the test) is small, then it is very likely that we will not reject the null even if it is wrong.
If we use the analogy with criminal trials, low power means that most likely the prosecution will not be able to provide sufficient evidence, even if the defendant is guilty.
Thus, in the case of lack of power, failure to reject is almost meaningless (it was anyway highly likely).
This is why, before performing a test, it is good statistical practice to compute its power against a relevant alternative .
If the power is found to be too small, there are usually remedies. In particular, statistical power can usually be increased by increasing the sample size (see, e.g., the lecture on hypothesis tests about the mean ).
As we have explained above, interpreting a failure to reject the null hypothesis is not always straightforward. Instead, interpreting a rejection is somewhat easier.
When we reject the null, we know that the data has provided a lot of evidence against the null. In other words, it is unlikely (how unlikely depends on the size of the test) that the null is true given the data we have observed.
There is an important caveat though. The null hypothesis is often made up of several assumptions, including:
the main assumption (the one we are testing);
other assumptions (e.g., technical assumptions) that we need to make in order to set up the hypothesis test.
For instance, in Example 2 above (reliability of a production plant), the main assumption is that the expected number of production halts per year is equal to 1. But there is also a technical assumption: the number of production halts has a Poisson distribution.
It must be kept in mind that a rejection is always a joint rejection of the main assumption and all the other assumptions .
Therefore, we should always ask ourselves whether the null has been rejected because the main assumption is wrong or because the other assumptions are violated.
In the case of Example 2 above, is a rejection of the null due to the fact that the expected number of halts is greater than 1 or is it due to the fact that the distribution of the number of halts is very different from a Poisson distribution?
When we suspect that a rejection is due to the inappropriateness of some technical assumption (e.g., assuming a Poisson distribution in the example), we say that the rejection could be due to misspecification of the model .
The right thing to do when these kind of suspicions arise is to conduct so-called robustness checks , that is, to change the technical assumptions and carry out the test again.
In our example, we could re-run the test by assuming a different probability distribution for the number of halts (e.g., a negative binomial or a compound Poisson - do not worry if you have never heard about these distributions).
If we keep obtaining a rejection of the null even after changing the technical assumptions several times, the we say that our rejection is robust to several different specifications of the model .
What are the main practical implications of everything we have said thus far? How does the theory above help us to set up and test a null hypothesis?
What we said can be summarized in the following guiding principles:
A test of hypothesis is like a criminal trial and you are the prosecutor . You want to find evidence that the defendant (the null hypothesis) is guilty. Your job is not to prove that the defendant is innocent. If you find yourself hoping that the defendant is found not guilty (i.e., the null is not rejected) then something is wrong with the way you set up the test. Remember: you are the prosecutor.
Compute the power of your test against one or more relevant alternative hypotheses. Do not run a test if you know ex-ante that it is unlikely to reject the null when the alternative hypothesis is true.
Beware of technical assumptions that you add to the main assumption you want to test. Make robustness checks in order to verify that the outcome of the test is not biased by model misspecification.
More examples of null hypotheses and how to test them can be found in the following lectures.
Where the example is found | Null hypothesis |
---|---|
The mean of a normal distribution is equal to a certain value | |
The variance of a normal distribution is equal to a certain value | |
A vector of parameters estimated by MLE satisfies a set of linear or non-linear restrictions | |
A regression coefficient is equal to a certain value |
The lecture on Hypothesis testing provides a more detailed mathematical treatment of null hypotheses and how they are tested.
This lecture on the null hypothesis was featured in Stanford University's Best practices in science .
Previous entry: Normal equations
Next entry: Parameter
Please cite as:
Taboga, Marco (2021). "Null hypothesis", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/null-hypothesis.
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On This Page:
A null hypothesis is a statistical concept suggesting no significant difference or relationship between measured variables. It’s the default assumption unless empirical evidence proves otherwise.
The null hypothesis states no relationship exists between the two variables being studied (i.e., one variable does not affect the other).
The null hypothesis is the statement that a researcher or an investigator wants to disprove.
Testing the null hypothesis can tell you whether your results are due to the effects of manipulating the dependent variable or due to random chance.
Null hypotheses (H0) start as research questions that the investigator rephrases as statements indicating no effect or relationship between the independent and dependent variables.
It is a default position that your research aims to challenge or confirm.
There is no significant difference in weight loss between individuals who exercise daily and those who do not.
Research Question | Null Hypothesis |
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Do teenagers use cell phones more than adults? | Teenagers and adults use cell phones the same amount. |
Do tomato plants exhibit a higher rate of growth when planted in compost rather than in soil? | Tomato plants show no difference in growth rates when planted in compost rather than soil. |
Does daily meditation decrease the incidence of depression? | Daily meditation does not decrease the incidence of depression. |
Does daily exercise increase test performance? | There is no relationship between daily exercise time and test performance. |
Does the new vaccine prevent infections? | The vaccine does not affect the infection rate. |
Does flossing your teeth affect the number of cavities? | Flossing your teeth has no effect on the number of cavities. |
We reject the null hypothesis when the data provide strong enough evidence to conclude that it is likely incorrect. This often occurs when the p-value (probability of observing the data given the null hypothesis is true) is below a predetermined significance level.
If the collected data does not meet the expectation of the null hypothesis, a researcher can conclude that the data lacks sufficient evidence to back up the null hypothesis, and thus the null hypothesis is rejected.
Rejecting the null hypothesis means that a relationship does exist between a set of variables and the effect is statistically significant ( p > 0.05).
If the data collected from the random sample is not statistically significance , then the null hypothesis will be accepted, and the researchers can conclude that there is no relationship between the variables.
You need to perform a statistical test on your data in order to evaluate how consistent it is with the null hypothesis. A p-value is one statistical measurement used to validate a hypothesis against observed data.
Calculating the p-value is a critical part of null-hypothesis significance testing because it quantifies how strongly the sample data contradicts the null hypothesis.
The level of statistical significance is often expressed as a p -value between 0 and 1. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.
Usually, a researcher uses a confidence level of 95% or 99% (p-value of 0.05 or 0.01) as general guidelines to decide if you should reject or keep the null.
When your p-value is less than or equal to your significance level, you reject the null hypothesis.
In other words, smaller p-values are taken as stronger evidence against the null hypothesis. Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis.
In this case, the sample data provides insufficient data to conclude that the effect exists in the population.
Because you can never know with complete certainty whether there is an effect in the population, your inferences about a population will sometimes be incorrect.
When you incorrectly reject the null hypothesis, it’s called a type I error. When you incorrectly fail to reject it, it’s called a type II error.
The reason we do not say “accept the null” is because we are always assuming the null hypothesis is true and then conducting a study to see if there is evidence against it. And, even if we don’t find evidence against it, a null hypothesis is not accepted.
A lack of evidence only means that you haven’t proven that something exists. It does not prove that something doesn’t exist.
It is risky to conclude that the null hypothesis is true merely because we did not find evidence to reject it. It is always possible that researchers elsewhere have disproved the null hypothesis, so we cannot accept it as true, but instead, we state that we failed to reject the null.
One can either reject the null hypothesis, or fail to reject it, but can never accept it.
We can never prove with 100% certainty that a hypothesis is true; We can only collect evidence that supports a theory. However, testing a hypothesis can set the stage for rejecting or accepting this hypothesis within a certain confidence level.
The null hypothesis is useful because it can tell us whether the results of our study are due to random chance or the manipulation of a variable (with a certain level of confidence).
A null hypothesis is rejected if the measured data is significantly unlikely to have occurred and a null hypothesis is accepted if the observed outcome is consistent with the position held by the null hypothesis.
Rejecting the null hypothesis sets the stage for further experimentation to see if a relationship between two variables exists.
Hypothesis testing is a critical part of the scientific method as it helps decide whether the results of a research study support a particular theory about a given population. Hypothesis testing is a systematic way of backing up researchers’ predictions with statistical analysis.
It helps provide sufficient statistical evidence that either favors or rejects a certain hypothesis about the population parameter.
The null (H0) and alternative (Ha or H1) hypotheses are two competing claims that describe the effect of the independent variable on the dependent variable. They are mutually exclusive, which means that only one of the two hypotheses can be true.
While the null hypothesis states that there is no effect in the population, an alternative hypothesis states that there is statistical significance between two variables.
The goal of hypothesis testing is to make inferences about a population based on a sample. In order to undertake hypothesis testing, you must express your research hypothesis as a null and alternative hypothesis. Both hypotheses are required to cover every possible outcome of the study.
The alternative hypothesis is the complement to the null hypothesis. The null hypothesis states that there is no effect or no relationship between variables, while the alternative hypothesis claims that there is an effect or relationship in the population.
It is the claim that you expect or hope will be true. The null hypothesis and the alternative hypothesis are always mutually exclusive, meaning that only one can be true at a time.
One major problem with the null hypothesis is that researchers typically will assume that accepting the null is a failure of the experiment. However, accepting or rejecting any hypothesis is a positive result. Even if the null is not refuted, the researchers will still learn something new.
We can either reject or fail to reject a null hypothesis, but never accept it. If your test fails to detect an effect, this is not proof that the effect doesn’t exist. It just means that your sample did not have enough evidence to conclude that it exists.
We can’t accept a null hypothesis because a lack of evidence does not prove something that does not exist. Instead, we fail to reject it.
Failing to reject the null indicates that the sample did not provide sufficient enough evidence to conclude that an effect exists.
If the p-value is greater than the significance level, then you fail to reject the null hypothesis.
A hypothesis test can either contain an alternative directional hypothesis or a non-directional alternative hypothesis. A directional hypothesis is one that contains the less than (“<“) or greater than (“>”) sign.
A nondirectional hypothesis contains the not equal sign (“≠”). However, a null hypothesis is neither directional nor non-directional.
A null hypothesis is a prediction that there will be no change, relationship, or difference between two variables.
The directional hypothesis or nondirectional hypothesis would then be considered alternative hypotheses to the null hypothesis.
Gill, J. (1999). The insignificance of null hypothesis significance testing. Political research quarterly , 52 (3), 647-674.
Krueger, J. (2001). Null hypothesis significance testing: On the survival of a flawed method. American Psychologist , 56 (1), 16.
Masson, M. E. (2011). A tutorial on a practical Bayesian alternative to null-hypothesis significance testing. Behavior research methods , 43 , 679-690.
Nickerson, R. S. (2000). Null hypothesis significance testing: a review of an old and continuing controversy. Psychological methods , 5 (2), 241.
Rozeboom, W. W. (1960). The fallacy of the null-hypothesis significance test. Psychological bulletin , 57 (5), 416.
Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.
A hypothesis is an assumption or idea, specifically a statistical claim about an unknown population parameter. For example, a judge assumes a person is innocent and verifies this by reviewing evidence and hearing testimony before reaching a verdict.
Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data.
To test the validity of the claim or assumption about the population parameter:
Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.
Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing.
One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.
There are two types of one-tailed test:
A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.
Example: H 0 : [Tex]\mu = [/Tex] 50 and H 1 : [Tex]\mu \neq 50 [/Tex]
To delve deeper into differences into both types of test: Refer to link
In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.
Null Hypothesis is True | Null Hypothesis is False | |
---|---|---|
Null Hypothesis is True (Accept) | Correct Decision | Type II Error (False Negative) |
Alternative Hypothesis is True (Reject) | Type I Error (False Positive) | Correct Decision |
Step 1: define null and alternative hypothesis.
State the null hypothesis ( [Tex]H_0 [/Tex] ), representing no effect, and the alternative hypothesis ( [Tex]H_1 [/Tex] ), suggesting an effect or difference.
We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.
Select a significance level ( [Tex]\alpha [/Tex] ), typically 0.05, to determine the threshold for rejecting the null hypothesis. It provides validity to our hypothesis test, ensuring that we have sufficient data to back up our claims. Usually, we determine our significance level beforehand of the test. The p-value is the criterion used to calculate our significance value.
Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.
The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.
There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.
We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.
T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.
In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.
Comparing the test statistic and tabulated critical value we have,
Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.
We can also come to an conclusion using the p-value,
Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.
At last, we can conclude our experiment using method A or B.
To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .
When population means and standard deviations are known.
[Tex]z = \frac{\bar{x} – \mu}{\frac{\sigma}{\sqrt{n}}}[/Tex]
T test is used when n<30,
t-statistic calculation is given by:
[Tex]t=\frac{x̄-μ}{s/\sqrt{n}} [/Tex]
Chi-Square Test for Independence categorical Data (Non-normally distributed) using:
[Tex]\chi^2 = \sum \frac{(O_{ij} – E_{ij})^2}{E_{ij}}[/Tex]
Let’s examine hypothesis testing using two real life situations,
Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.
Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.
If the evidence suggests less than a 5% chance of observing the results due to random variation.
Using paired T-test analyze the data to obtain a test statistic and a p-value.
The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.
t = m/(s/√n)
then, m= -3.9, s= 1.8 and n= 10
we, calculate the , T-statistic = -9 based on the formula for paired t test
The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.
thus, p-value = 8.538051223166285e-06
Step 5: Result
Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.
Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.
Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.
We will implement our first real life problem via python,
import numpy as np from scipy import stats # Data before_treatment = np . array ([ 120 , 122 , 118 , 130 , 125 , 128 , 115 , 121 , 123 , 119 ]) after_treatment = np . array ([ 115 , 120 , 112 , 128 , 122 , 125 , 110 , 117 , 119 , 114 ]) # Step 1: Null and Alternate Hypotheses # Null Hypothesis: The new drug has no effect on blood pressure. # Alternate Hypothesis: The new drug has an effect on blood pressure. null_hypothesis = "The new drug has no effect on blood pressure." alternate_hypothesis = "The new drug has an effect on blood pressure." # Step 2: Significance Level alpha = 0.05 # Step 3: Paired T-test t_statistic , p_value = stats . ttest_rel ( after_treatment , before_treatment ) # Step 4: Calculate T-statistic manually m = np . mean ( after_treatment - before_treatment ) s = np . std ( after_treatment - before_treatment , ddof = 1 ) # using ddof=1 for sample standard deviation n = len ( before_treatment ) t_statistic_manual = m / ( s / np . sqrt ( n )) # Step 5: Decision if p_value <= alpha : decision = "Reject" else : decision = "Fail to reject" # Conclusion if decision == "Reject" : conclusion = "There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different." else : conclusion = "There is insufficient evidence to claim a significant difference in average blood pressure before and after treatment with the new drug." # Display results print ( "T-statistic (from scipy):" , t_statistic ) print ( "P-value (from scipy):" , p_value ) print ( "T-statistic (calculated manually):" , t_statistic_manual ) print ( f "Decision: { decision } the null hypothesis at alpha= { alpha } ." ) print ( "Conclusion:" , conclusion )
T-statistic (from scipy): -9.0 P-value (from scipy): 8.538051223166285e-06 T-statistic (calculated manually): -9.0 Decision: Reject the null hypothesis at alpha=0.05. Conclusion: There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.
In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05.
Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.
Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.
Populations Mean = 200
Population Standard Deviation (σ): 5 mg/dL(given for this problem)
As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.
The test statistic is calculated by using the z formula Z = [Tex](203.8 – 200) / (5 \div \sqrt{25}) [/Tex] and we get accordingly , Z =2.039999999999992.
Step 4: Result
Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL
import scipy.stats as stats import math import numpy as np # Given data sample_data = np . array ( [ 205 , 198 , 210 , 190 , 215 , 205 , 200 , 192 , 198 , 205 , 198 , 202 , 208 , 200 , 205 , 198 , 205 , 210 , 192 , 205 , 198 , 205 , 210 , 192 , 205 ]) population_std_dev = 5 population_mean = 200 sample_size = len ( sample_data ) # Step 1: Define the Hypotheses # Null Hypothesis (H0): The average cholesterol level in a population is 200 mg/dL. # Alternate Hypothesis (H1): The average cholesterol level in a population is different from 200 mg/dL. # Step 2: Define the Significance Level alpha = 0.05 # Two-tailed test # Critical values for a significance level of 0.05 (two-tailed) critical_value_left = stats . norm . ppf ( alpha / 2 ) critical_value_right = - critical_value_left # Step 3: Compute the test statistic sample_mean = sample_data . mean () z_score = ( sample_mean - population_mean ) / \ ( population_std_dev / math . sqrt ( sample_size )) # Step 4: Result # Check if the absolute value of the test statistic is greater than the critical values if abs ( z_score ) > max ( abs ( critical_value_left ), abs ( critical_value_right )): print ( "Reject the null hypothesis." ) print ( "There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL." ) else : print ( "Fail to reject the null hypothesis." ) print ( "There is not enough evidence to conclude that the average cholesterol level in the population is different from 200 mg/dL." )
Reject the null hypothesis. There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL.
Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.
1. what are the 3 types of hypothesis test.
There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.
Null Hypothesis ( [Tex]H_o [/Tex] ): No effect or difference exists. Alternative Hypothesis ( [Tex]H_1 [/Tex] ): An effect or difference exists. Significance Level ( [Tex]\alpha [/Tex] ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.
Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.
Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.
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Null Hypothesis H 0: The correlation in the population is zero: ρ = 0. Alternative Hypothesis H A: The correlation in the population is not zero: ρ ≠ 0. For all these cases, the analysts define the hypotheses before the study. After collecting the data, they perform a hypothesis test to determine whether they can reject the null hypothesis.
When using the p-value to evaluate a hypothesis test, the following rhymes can come in handy:. If the p-value is low, the null must go.. If the p-value is high, the null must fly.. This memory aid relates a p-value less than the established alpha ("the p-value is low") as rejecting the null hypothesis and, likewise, relates a p-value higher than the established alpha ("the p-value is ...
The null hypothesis (H0) answers "No, there's no effect in the population.". The alternative hypothesis (Ha) answers "Yes, there is an effect in the population.". The null and alternative are always claims about the population. That's because the goal of hypothesis testing is to make inferences about a population based on a sample.
The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. \(H_0\): The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.
Hypothesis Testing | A Step-by-Step Guide with Easy ...
The null hypothesis and the alternative hypothesis are types of conjectures used in statistical tests to make statistical inferences, which are formal methods of reaching conclusions and separating scientific claims from statistical noise.. The statement being tested in a test of statistical significance is called the null hypothesis. The test of significance is designed to assess the strength ...
The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. H 0, the —null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.
In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted \(H_0\) while the alternative hypothesis is usually denoted \(H_1\). An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ...
The Null hypothesis \(\left(H_{O}\right)\) is a statement about the comparisons, e.g., between a sample statistic and the population, or between two treatment groups. The former is referred to as a one-tailed test whereas the latter is called a two-tailed test. The null hypothesis is typically "no statistical difference" between the ...
A crucial step in null hypothesis testing is finding the likelihood of the sample result if the null hypothesis were true. This probability is called the p value. A low p value means that the sample result would be unlikely if the null hypothesis were true and leads to the rejection of the null hypothesis. A high p value means that the sample ...
When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H 0 and the decision to reject or not. The outcomes are summarized in the following table: Figure 5.14: Type I and type II errors; H 0 IS ACTUALLY; Action: True:
S.3 Hypothesis Testing. In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail. The general idea of hypothesis testing involves: Making an initial assumption. Collecting evidence (data).
An example of Neyman-Pearson hypothesis testing (or null hypothesis statistical significance testing) can be made by a change to the radioactive suitcase example. If the "suitcase" is actually a shielded container for the transportation of radioactive material, then a test might be used to select among three hypotheses: no radioactive source ...
A hypothesis test uses sample data to determine whether or not some claim about a population parameter is true. Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms: H 0 (Null Hypothesis): Population parameter =, ≤, ≥ some value
Null and Alternative Hypotheses | Introduction to Statistics
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.
Let's return finally to the question of whether we reject or fail to reject the null hypothesis. If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above ...
A hypothesis test consists of five steps: 1. State the hypotheses. State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false. 2. Determine a significance level to use for the hypothesis. Decide on a significance level.
In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis, typically denoted with H0.
Step 1: Figure out the hypothesis from the problem. The hypothesis is usually hidden in a word problem, and is sometimes a statement of what you expect to happen in the experiment. The hypothesis in the above question is "I expect the average recovery period to be greater than 8.2 weeks.". Step 2: Convert the hypothesis to math.
Null hypothesis. by Marco Taboga, PhD. In a test of hypothesis, a sample of data is used to decide whether to reject or not to reject a hypothesis about the probability distribution from which the sample was extracted.. The hypothesis is called the null hypothesis, or simply "the null".
A tutorial on a practical Bayesian alternative to null-hypothesis significance testing. Behavior research methods, 43, 679-690. Nickerson, R. S. (2000). Null hypothesis significance testing: a review of an old and continuing controversy. Psychological methods, 5(2), 241. Rozeboom, W. W. (1960). The fallacy of the null-hypothesis significance test.
Aug 5, 2022. 6. Photo by Andrew George on Unsplash. Student's t-tests are commonly used in inferential statistics for testing a hypothesis on the basis of a difference between sample means. However, people often misinterpret the results of t-tests, which leads to false research findings and a lack of reproducibility of studies.
Null hypothesis, often denoted as H0, is a foundational concept in statistical hypothesis testing. It represents an assumption that no significant difference, effect, or relationship exists between variables within a population. Learn more about Null Hypothesis, its formula, symbol and example in this article
16.3.5 Step 5: Determine the probability of the data under the null hypothesis. This is the step where NHST starts to violate our intuition - rather than determining the likelihood that the null hypothesis is true given the data, we instead determine the likelihood of the data under the null hypothesis - because we started out by assuming that the null hypothesis is true!
Test Statistic: The test statistic is a numerical value calculated from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It is compared to a critical value or p-value to make decisions about the statistical significance of the observed results.
Statistics document from University of Cincinnati, Main Campus, 1 page, CHAPTER 8 Hypothesis test- inferential procedure; uses data from sample to draw conclusion about population Critical region- sample outcomes that are very UNLIKELY to occur if null is true. Enough to reject hypothesis Alpha level- probability of type 1 er