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A statistical hypothesis is an assumption about a population parameter .
For example, we may assume that the mean height of a male in the U.S. is 70 inches.
The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter .
A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.
To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data.
There are two types of statistical hypotheses:
The null hypothesis , denoted as H 0 , is the hypothesis that the sample data occurs purely from chance.
The alternative hypothesis , denoted as H 1 or H a , is the hypothesis that the sample data is influenced by some non-random cause.
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. Common choices are .01, .05, and .1.
3. Find the test statistic.
Find the test statistic and the corresponding p-value. Often we are analyzing a population mean or proportion and the general formula to find the test statistic is: (sample statistic – population parameter) / (standard deviation of statistic)
4. Reject or fail to reject the null hypothesis.
Using the test statistic or the p-value, determine if you can reject or fail to reject the null hypothesis based on the significance level.
The p-value tells us the strength of evidence in support of a null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis.
5. Interpret the results.
Interpret the results of the hypothesis test in the context of the question being asked.
There are two types of decision errors that one can make when doing a hypothesis test:
Type I error: You reject the null hypothesis when it is actually true. The probability of committing a Type I error is equal to the significance level, often called alpha , and denoted as α.
Type II error: You fail to reject the null hypothesis when it is actually false. The probability of committing a Type II error is called the Power of the test or Beta , denoted as β.
A statistical hypothesis can be one-tailed or two-tailed.
A one-tailed hypothesis involves making a “greater than” or “less than ” statement.
For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches. The null hypothesis would be H0: µ ≥ 70 inches and the alternative hypothesis would be Ha: µ < 70 inches.
A two-tailed hypothesis involves making an “equal to” or “not equal to” statement.
For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null hypothesis would be H0: µ = 70 inches and the alternative hypothesis would be Ha: µ ≠ 70 inches.
Note: The “equal” sign is always included in the null hypothesis, whether it is =, ≥, or ≤.
Related: What is a Directional Hypothesis?
There are many different types of hypothesis tests you can perform depending on the type of data you’re working with and the goal of your analysis.
The following tutorials provide an explanation of the most common types of hypothesis tests:
Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test Introduction to the One Proportion Z-Test Introduction to the Two Proportion Z-Test
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Priya ranganathan.
1 Department of Anesthesiology, Critical Care and Pain, Tata Memorial Hospital, Mumbai, Maharashtra, India
2 Department of Surgical Oncology, Tata Memorial Centre, Mumbai, Maharashtra, India
The second article in this series on biostatistics covers the concepts of sample, population, research hypotheses and statistical errors.
Ranganathan P, Pramesh CS. An Introduction to Statistics: Understanding Hypothesis Testing and Statistical Errors. Indian J Crit Care Med 2019;23(Suppl 3):S230–S231.
Two papers quoted in this issue of the Indian Journal of Critical Care Medicine report. The results of studies aim to prove that a new intervention is better than (superior to) an existing treatment. In the ABLE study, the investigators wanted to show that transfusion of fresh red blood cells would be superior to standard-issue red cells in reducing 90-day mortality in ICU patients. 1 The PROPPR study was designed to prove that transfusion of a lower ratio of plasma and platelets to red cells would be superior to a higher ratio in decreasing 24-hour and 30-day mortality in critically ill patients. 2 These studies are known as superiority studies (as opposed to noninferiority or equivalence studies which will be discussed in a subsequent article).
A sample represents a group of participants selected from the entire population. Since studies cannot be carried out on entire populations, researchers choose samples, which are representative of the population. This is similar to walking into a grocery store and examining a few grains of rice or wheat before purchasing an entire bag; we assume that the few grains that we select (the sample) are representative of the entire sack of grains (the population).
The results of the study are then extrapolated to generate inferences about the population. We do this using a process known as hypothesis testing. This means that the results of the study may not always be identical to the results we would expect to find in the population; i.e., there is the possibility that the study results may be erroneous.
A clinical trial begins with an assumption or belief, and then proceeds to either prove or disprove this assumption. In statistical terms, this belief or assumption is known as a hypothesis. Counterintuitively, what the researcher believes in (or is trying to prove) is called the “alternate” hypothesis, and the opposite is called the “null” hypothesis; every study has a null hypothesis and an alternate hypothesis. For superiority studies, the alternate hypothesis states that one treatment (usually the new or experimental treatment) is superior to the other; the null hypothesis states that there is no difference between the treatments (the treatments are equal). For example, in the ABLE study, we start by stating the null hypothesis—there is no difference in mortality between groups receiving fresh RBCs and standard-issue RBCs. We then state the alternate hypothesis—There is a difference between groups receiving fresh RBCs and standard-issue RBCs. It is important to note that we have stated that the groups are different, without specifying which group will be better than the other. This is known as a two-tailed hypothesis and it allows us to test for superiority on either side (using a two-sided test). This is because, when we start a study, we are not 100% certain that the new treatment can only be better than the standard treatment—it could be worse, and if it is so, the study should pick it up as well. One tailed hypothesis and one-sided statistical testing is done for non-inferiority studies, which will be discussed in a subsequent paper in this series.
There are two possibilities to consider when interpreting the results of a superiority study. The first possibility is that there is truly no difference between the treatments but the study finds that they are different. This is called a Type-1 error or false-positive error or alpha error. This means falsely rejecting the null hypothesis.
The second possibility is that there is a difference between the treatments and the study does not pick up this difference. This is called a Type 2 error or false-negative error or beta error. This means falsely accepting the null hypothesis.
The power of the study is the ability to detect a difference between groups and is the converse of the beta error; i.e., power = 1-beta error. Alpha and beta errors are finalized when the protocol is written and form the basis for sample size calculation for the study. In an ideal world, we would not like any error in the results of our study; however, we would need to do the study in the entire population (infinite sample size) to be able to get a 0% alpha and beta error. These two errors enable us to do studies with realistic sample sizes, with the compromise that there is a small possibility that the results may not always reflect the truth. The basis for this will be discussed in a subsequent paper in this series dealing with sample size calculation.
Conventionally, type 1 or alpha error is set at 5%. This means, that at the end of the study, if there is a difference between groups, we want to be 95% certain that this is a true difference and allow only a 5% probability that this difference has occurred by chance (false positive). Type 2 or beta error is usually set between 10% and 20%; therefore, the power of the study is 90% or 80%. This means that if there is a difference between groups, we want to be 80% (or 90%) certain that the study will detect that difference. For example, in the ABLE study, sample size was calculated with a type 1 error of 5% (two-sided) and power of 90% (type 2 error of 10%) (1).
Table 1 gives a summary of the two types of statistical errors with an example
Statistical errors
(a) Types of statistical errors | |||
: Null hypothesis is | |||
True | False | ||
Null hypothesis is actually | True | Correct results! | Falsely rejecting null hypothesis - Type I error |
False | Falsely accepting null hypothesis - Type II error | Correct results! | |
(b) Possible statistical errors in the ABLE trial | |||
There is difference in mortality between groups receiving fresh RBCs and standard-issue RBCs | There difference in mortality between groups receiving fresh RBCs and standard-issue RBCs | ||
Truth | There is difference in mortality between groups receiving fresh RBCs and standard-issue RBCs | Correct results! | Falsely rejecting null hypothesis - Type I error |
There difference in mortality between groups receiving fresh RBCs and standard-issue RBCs | Falsely accepting null hypothesis - Type II error | Correct results! |
In the next article in this series, we will look at the meaning and interpretation of ‘ p ’ value and confidence intervals for hypothesis testing.
Source of support: Nil
Conflict of interest: None
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!
Content preview.
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6a.2 - steps for hypothesis tests, the logic of hypothesis testing section .
A hypothesis, in statistics, is a statement about a population parameter, where this statement typically is represented by some specific numerical value. In testing a hypothesis, we use a method where we gather data in an effort to gather evidence about the hypothesis.
How do we decide whether to reject the null hypothesis?
In hypothesis testing, there are certain steps one must follow. Below these are summarized into six such steps to conducting a test of a hypothesis.
We will follow these six steps for the remainder of this Lesson. In the future Lessons, the steps will be followed but may not be explained explicitly.
Step 1 is a very important step to set up correctly. If your hypotheses are incorrect, your conclusion will be incorrect. In this next section, we practice with Step 1 for the one sample situations.
Descriptive statistics, inferential statistics, stat reference, statistics - hypothesis testing.
Hypothesis testing is a formal way of checking if a hypothesis about a population is true or not.
A hypothesis is a claim about a population parameter .
A hypothesis test is a formal procedure to check if a hypothesis is true or not.
Examples of claims that can be checked:
The average height of people in Denmark is more than 170 cm.
The share of left handed people in Australia is not 10%.
The average income of dentists is less the average income of lawyers.
Hypothesis testing is based on making two different claims about a population parameter.
The null hypothesis (\(H_{0} \)) and the alternative hypothesis (\(H_{1}\)) are the claims.
The two claims needs to be mutually exclusive , meaning only one of them can be true.
The alternative hypothesis is typically what we are trying to prove.
For example, we want to check the following claim:
"The average height of people in Denmark is more than 170 cm."
In this case, the parameter is the average height of people in Denmark (\(\mu\)).
The null and alternative hypothesis would be:
Null hypothesis : The average height of people in Denmark is 170 cm.
Alternative hypothesis : The average height of people in Denmark is more than 170 cm.
The claims are often expressed with symbols like this:
\(H_{0}\): \(\mu = 170 \: cm \)
\(H_{1}\): \(\mu > 170 \: cm \)
If the data supports the alternative hypothesis, we reject the null hypothesis and accept the alternative hypothesis.
If the data does not support the alternative hypothesis, we keep the null hypothesis.
Note: The alternative hypothesis is also referred to as (\(H_{A} \)).
The significance level (\(\alpha\)) is the uncertainty we accept when rejecting the null hypothesis in the hypothesis test.
The significance level is a percentage probability of accidentally making the wrong conclusion.
Typical significance levels are:
A lower significance level means that the evidence in the data needs to be stronger to reject the null hypothesis.
There is no "correct" significance level - it only states the uncertainty of the conclusion.
Note: A 5% significance level means that when we reject a null hypothesis:
We expect to reject a true null hypothesis 5 out of 100 times.
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The test statistic is used to decide the outcome of the hypothesis test.
The test statistic is a standardized value calculated from the sample.
Standardization means converting a statistic to a well known probability distribution .
The type of probability distribution depends on the type of test.
Common examples are:
Note: You will learn how to calculate the test statistic for each type of test in the following chapters.
There are two main approaches used for hypothesis tests:
The critical value approach checks if the test statistic is in the rejection region .
The rejection region is an area of probability in the tails of the distribution.
The size of the rejection region is decided by the significance level (\(\alpha\)).
The value that separates the rejection region from the rest is called the critical value .
Here is a graphical illustration:
If the test statistic is inside this rejection region, the null hypothesis is rejected .
For example, if the test statistic is 2.3 and the critical value is 2 for a significance level (\(\alpha = 0.05\)):
We reject the null hypothesis (\(H_{0} \)) at 0.05 significance level (\(\alpha\))
The p-value approach checks if the p-value of the test statistic is smaller than the significance level (\(\alpha\)).
The p-value of the test statistic is the area of probability in the tails of the distribution from the value of the test statistic.
If the p-value is smaller than the significance level, the null hypothesis is rejected .
The p-value directly tells us the lowest significance level where we can reject the null hypothesis.
For example, if the p-value is 0.03:
We reject the null hypothesis (\(H_{0} \)) at a 0.05 significance level (\(\alpha\))
We keep the null hypothesis (\(H_{0}\)) at a 0.01 significance level (\(\alpha\))
Note: The two approaches are only different in how they present the conclusion.
The following steps are used for a hypothesis test:
One condition is that the sample is randomly selected from the population.
The other conditions depends on what type of parameter you are testing the hypothesis for.
Common parameters to test hypotheses are:
You will learn the steps for both types in the following pages.
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You will learn the essentials of hypothesis tests, from fundamental concepts to practical applications in statistics.
Hypothesis testing is a statistical tool used to make decisions based on data.
It involves making assumptions about a population parameter and testing its validity using a population sample.
Hypothesis tests help us draw conclusions and make informed decisions in various fields like business, research, and science.
The null hypothesis (H0) is an initial claim about a population parameter, typically representing no effect or no difference.
The alternative hypothesis (H1) opposes the null hypothesis, suggesting an effect or difference.
Hypothesis tests aim to determine if there is evidence for the null hypothesis rejection in favor of the alternative hypothesis.
The significance level (α), often set at 0.05 or 5%, serves as a threshold for determining if we should reject the null hypothesis.
A p-value, calculated during hypothesis testing, represents the probability of observing the test statistic if the null hypothesis is true.
Suppose the p-value is less than the significance level. We reject the null hypothesis, in that case, indicating that the alternative hypothesis is more likely.
Parametric tests assume the data follows a specific probability distribution, usually the normal distribution. Examples include the Student’s t-test.
Non-parametric tests do not require such assumptions and are helpful when dealing with data that do not meet the assumptions of parametric tests. Examples include the Mann-Whitney U test.
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Independent samples t-test: This analysis compares the means of two independent groups.
Paired samples t-test: Compares the means of two related groups (e.g., before and after treatment).
Chi-squared test: Determines if there is a significant association, in a contingency table, between two categorical variables.
Analysis of Variance (ANOVA): Compares the means of three or more independent groups to determine whether significant differences exist.
Pearson’s Correlation Coefficient (Pearson’s r): Quantifies the strength and direction of a linear association between two continuous variables.
Simple Linear Regression: Evaluate whether a significant linear relationship exists between a predictor variable (X) and a continuous outcome variable (y).
Logistic Regression: Determines the relationship between one or more predictor variables (continuous or categorical) and a binary outcome variable (e.g., success or failure).
Levene’s Test: Tests the equality of variances between two or more groups, often used as an assumption checks for ANOVA.
Shapiro-Wilk Test: Assesses the null hypothesis that a data sample is drawn from a population with a normal distribution.
Hypothesis Test | Description | Application |
---|---|---|
Compares means of two independent groups | Comparing scores of two groups of students | |
Compares means of two related groups (e.g., before and after treatment) | Comparing weight loss before and after a diet program | |
Determines significant associations between two categorical variables in a contingency table | Analyzing the relationship between education and income | |
Compares means of three or more independent groups | Evaluating the impact of different teaching methods on test scores | |
Measures the strength and direction of a linear relationship between two continuous variables | Studying the correlation between height and weight | |
Determines a significant linear relationship between a predictor variable and an outcome variable | Predicting sales based on advertising budget | |
Determines the relationship between predictor variables and a binary outcome variable | Predicting the probability of loan default based on credit score | |
Tests the equality of variances between two or more groups | Checking the assumption of equal variances for ANOVA | |
Tests if a data sample is from a normally distributed population | Assessing normality assumption for parametric tests |
To interpret the hypothesis test results, compare the p-value to the chosen significance level.
If the p-value falls below the significance level, reject the null hypothesis and infer that a notable effect or difference exists.
Otherwise, fail to reject the null hypothesis, meaning there is insufficient evidence to support the alternative hypothesis.
In addition to understanding the basics of hypothesis tests, it’s crucial to consider other relevant information when interpreting the results.
For example, factors such as effect size, statistical power, and confidence intervals can provide valuable insights and help you make more informed decisions.
Effect size
The effect size represents a quantitative measurement of the strength or magnitude of the observed relationship or effect between variables. It aids in evaluating the practical significance of the results. A statistically significant outcome may not necessarily imply practical relevance. At the same time, a substantial effect size can suggest meaningful findings, even when statistical significance appears marginal.
Statistical power
The power of a test represents the likelihood of accurately rejecting the null hypothesis when it is incorrect. In other words, it’s the likelihood that the test will detect an effect when it exists. Factors affecting the power of a test include the sample size, effect size, and significance level. Enhanced power reduces the likelihood of making an error of Type II — failing to reject the null hypothesis when it ought to be rejected.
Confidence intervals
A confidence interval represents a range where the true population parameter is expected to be found with a specified confidence level (e.g., 95%). Confidence intervals provide additional context to hypothesis testing, helping to assess the estimate’s precision and offering a better understanding of the uncertainty surrounding the results.
By considering these additional aspects when interpreting the results of hypothesis tests, you can gain a more comprehensive understanding of the data and make more informed conclusions.
Hypothesis testing is an indispensable statistical tool for drawing meaningful inferences and making informed data-based decisions.
By comprehending the essential concepts such as null and alternative hypotheses, significance levels, p-values, and the distinction between parametric and non-parametric tests, you can proficiently apply hypothesis testing to a wide range of real-world situations.
Additionally, understanding the importance of effect sizes, statistical power, and confidence intervals will enhance your ability to interpret the results and make better decisions.
With many applications across various fields, including medicine, psychology, business, and environmental sciences, hypothesis testing is a versatile and valuable method for research and data analysis.
A comprehensive grasp of hypothesis testing techniques will enable professionals and researchers to strengthen their decision-making processes, optimize strategies, and deepen their understanding of the relationships between variables, leading to more impactful results and discoveries.
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Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.
Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data.
Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.
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.
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.
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.
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.
T test is used when n<30,
t-statistic calculation is given by:
Chi-Square Test for Independence categorical Data (Non-normally distributed) using:
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,
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.
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
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 ( ): No effect or difference exists. Alternative Hypothesis ( ): An effect or difference exists. Significance Level ( ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.
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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Published on September 4, 2020 by Pritha Bhandari . Revised on June 22, 2023.
While descriptive statistics summarize the characteristics of a data set, inferential statistics help you come to conclusions and make predictions based on your data.
When you have collected data from a sample , you can use inferential statistics to understand the larger population from which the sample is taken.
Inferential statistics have two main uses:
Descriptive versus inferential statistics, estimating population parameters from sample statistics, hypothesis testing, other interesting articles, frequently asked questions about inferential statistics.
Descriptive statistics allow you to describe a data set, while inferential statistics allow you to make inferences based on a data set.
Using descriptive statistics, you can report characteristics of your data:
In descriptive statistics, there is no uncertainty – the statistics precisely describe the data that you collected. If you collect data from an entire population, you can directly compare these descriptive statistics to those from other populations.
Most of the time, you can only acquire data from samples, because it is too difficult or expensive to collect data from the whole population that you’re interested in.
While descriptive statistics can only summarize a sample’s characteristics, inferential statistics use your sample to make reasonable guesses about the larger population.
With inferential statistics, it’s important to use random and unbiased sampling methods . If your sample isn’t representative of your population, then you can’t make valid statistical inferences or generalize .
Since the size of a sample is always smaller than the size of the population, some of the population isn’t captured by sample data. This creates sampling error , which is the difference between the true population values (called parameters) and the measured sample values (called statistics).
Sampling error arises any time you use a sample, even if your sample is random and unbiased. For this reason, there is always some uncertainty in inferential statistics. However, using probability sampling methods reduces this uncertainty.
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The characteristics of samples and populations are described by numbers called statistics and parameters :
Sampling error is the difference between a parameter and a corresponding statistic. Since in most cases you don’t know the real population parameter, you can use inferential statistics to estimate these parameters in a way that takes sampling error into account.
There are two important types of estimates you can make about the population: point estimates and interval estimates .
Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie.
A confidence interval uses the variability around a statistic to come up with an interval estimate for a parameter. Confidence intervals are useful for estimating parameters because they take sampling error into account.
While a point estimate gives you a precise value for the parameter you are interested in, a confidence interval tells you the uncertainty of the point estimate. They are best used in combination with each other.
Each confidence interval is associated with a confidence level. A confidence level tells you the probability (in percentage) of the interval containing the parameter estimate if you repeat the study again.
A 95% confidence interval means that if you repeat your study with a new sample in exactly the same way 100 times, you can expect your estimate to lie within the specified range of values 95 times.
Although you can say that your estimate will lie within the interval a certain percentage of the time, you cannot say for sure that the actual population parameter will. That’s because you can’t know the true value of the population parameter without collecting data from the full population.
However, with random sampling and a suitable sample size, you can reasonably expect your confidence interval to contain the parameter a certain percentage of the time.
Your point estimate of the population mean paid vacation days is the sample mean of 19 paid vacation days.
Hypothesis testing is a formal process of statistical analysis using inferential statistics. The goal of hypothesis testing is to compare populations or assess relationships between variables using samples.
Hypotheses , or predictions, are tested using statistical tests . Statistical tests also estimate sampling errors so that valid inferences can be made.
Statistical tests can be parametric or non-parametric. Parametric tests are considered more statistically powerful because they are more likely to detect an effect if one exists.
Parametric tests make assumptions that include the following:
When your data violates any of these assumptions, non-parametric tests are more suitable. Non-parametric tests are called “distribution-free tests” because they don’t assume anything about the distribution of the population data.
Statistical tests come in three forms: tests of comparison, correlation or regression.
Comparison tests assess whether there are differences in means, medians or rankings of scores of two or more groups.
To decide which test suits your aim, consider whether your data meets the conditions necessary for parametric tests, the number of samples, and the levels of measurement of your variables.
Means can only be found for interval or ratio data , while medians and rankings are more appropriate measures for ordinal data .
test | Yes | Means | 2 samples |
---|---|---|---|
Yes | Means | 3+ samples | |
Mood’s median | No | Medians | 2+ samples |
Wilcoxon signed-rank | No | Distributions | 2 samples |
Wilcoxon rank-sum (Mann-Whitney ) | No | Sums of rankings | 2 samples |
Kruskal-Wallis | No | Mean rankings | 3+ samples |
Correlation tests determine the extent to which two variables are associated.
Although Pearson’s r is the most statistically powerful test, Spearman’s r is appropriate for interval and ratio variables when the data doesn’t follow a normal distribution.
The chi square test of independence is the only test that can be used with nominal variables.
Pearson’s | Yes | Interval/ratio variables |
---|---|---|
Spearman’s | No | Ordinal/interval/ratio variables |
Chi square test of independence | No | Nominal/ordinal variables |
Regression tests demonstrate whether changes in predictor variables cause changes in an outcome variable. You can decide which regression test to use based on the number and types of variables you have as predictors and outcomes.
Most of the commonly used regression tests are parametric. If your data is not normally distributed, you can perform data transformations.
Data transformations help you make your data normally distributed using mathematical operations, like taking the square root of each value.
1 interval/ratio variable | 1 interval/ratio variable | |
2+ interval/ratio variable(s) | 1 interval/ratio variable | |
Logistic regression | 1+ any variable(s) | 1 binary variable |
Nominal regression | 1+ any variable(s) | 1 nominal variable |
Ordinal regression | 1+ any variable(s) | 1 ordinal variable |
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
Methodology
Research bias
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Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.
A statistic refers to measures about the sample , while a parameter refers to measures about the population .
A sampling error is the difference between a population parameter and a sample statistic .
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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Bhandari, P. (2023, June 22). Inferential Statistics | An Easy Introduction & Examples. Scribbr. Retrieved June 18, 2024, from https://www.scribbr.com/statistics/inferential-statistics/
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The bottom line.
Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.
Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population or a data-generating process. The word "population" will be used for both of these cases in the following descriptions.
In hypothesis testing, an analyst tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.
The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis. Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.
The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.
If an individual wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct. Mathematically, the null hypothesis is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.
A random sample of 100 coin flips is taken, and the null hypothesis is tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.
If there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."
Some statisticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”
Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.
Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.
Hypothesis testing refers to a statistical process that helps researchers determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. All hypothesis testing methods have the same four-step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.
Sage. " Introduction to Hypothesis Testing ," Page 4.
Elder Research. " Who Invented the Null Hypothesis? "
Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples ."
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We will cover the seven steps one by one.
The null hypothesis can be thought of as the opposite of the "guess" the researchers made: in this example, the biologist thinks the plant height will be different for the fertilizers. So the null would be that there will be no difference among the groups of plants. Specifically, in more statistical language the null for an ANOVA is that the means are the same. We state the null hypothesis as: \[H_{0}: \ \mu_{1} = \mu_{2} = \ldots = \mu_{T}\] for \(T\) levels of an experimental treatment.
Why do we do this? Why not simply test the working hypothesis directly? The answer lies in the Popperian Principle of Falsification. Karl Popper (a philosopher) discovered that we can't conclusively confirm a hypothesis, but we can conclusively negate one. So we set up a null hypothesis which is effectively the opposite of the working hypothesis. The hope is that based on the strength of the data, we will be able to negate or reject the null hypothesis and accept an alternative hypothesis. In other words, we usually see the working hypothesis in \(H_{A}\).
\[H_{A}: \ \text{treatment level means not all equal}\]
The reason we state the alternative hypothesis this way is that if the null is rejected, there are many possibilities.
For example, \(\mu_{1} \neq \mu_{2} = \ldots = \mu_{T}\) is one possibility, as is \(\mu_{1} = \mu_{2} \neq \mu_{3} = \ldots = \mu_{T}\). Many people make the mistake of stating the alternative hypothesis as \(mu_{1} \neq mu_{2} \neq \ldots \neq \mu_{T}\), which says that every mean differs from every other mean. This is a possibility, but only one of many possibilities. To cover all alternative outcomes, we resort to a verbal statement of "not all equal" and then follow up with mean comparisons to find out where differences among means exist. In our example, this means that fertilizer 1 may result in plants that are really tall, but fertilizers 2, 3, and the plants with no fertilizers don't differ from one another. A simpler way of thinking about this is that at least one mean is different from all others.
If we look at what can happen in a hypothesis test, we can construct the following contingency table:
\(H_{0}\) is TRUE | \(H_{0}\) is FALSE | |
Accept \(H_{0}\) | correct | Type II Error \(\beta\) = probability of Type II Error |
Reject \(H_{0}\) | Type I Error | correct |
You should be familiar with type I and type II errors from your introductory course. It is important to note that we want to set \(\alpha\) before the experiment ( a priori ) because the Type I error is the more grievous error to make. The typical value of \(\alpha\) is 0.05, establishing a 95% confidence level. For this course, we will assume \(\alpha\) =0.05, unless stated otherwise.
Remember the importance of recognizing whether data is collected through an experimental design or observational study.
For categorical treatment level means, we use an \(F\) statistic, named after R.A. Fisher. We will explore the mechanics of computing the \(F\) statistic beginning in Chapter 2. The \(F\) value we get from the data is labeled \(F_{\text{calculated}}\).
As with all other test statistics, a threshold (critical) value of \(F\) is established. This \(F\) value can be obtained from statistical tables or software and is referred to as \(F_{\text{critical}}\) or \(F_{\alpha}\). As a reminder, this critical value is the minimum value for the test statistic (in this case the F test) for us to be able to reject the null.
The \(F\) distribution, \(F_{\alpha}\), and the location of acceptance and rejection regions are shown in the graph below:
If the \(F_{\text{\calculated}}\) from the data is larger than the \(F_{\alpha}\), then you are in the rejection region and you can reject the null hypothesis with \((1 - \alpha)\) level of confidence.
Note that modern statistical software condenses steps 6 and 7 by providing a \(p\)-value. The \(p\)-value here is the probability of getting an \(F_{\text{calculated}}\) even greater than what you observe assuming the null hypothesis is true. If by chance, the \(F_{\text{calculated}} = F_{\alpha}\), then the \(p\)-value would exactly equal \(\alpha\). With larger \(F_{\text{calculated}}\) values, we move further into the rejection region and the \(p\) - value becomes less than \(\alpha\). So the decision rule is as follows:
If the \(p\) - value obtained from the ANOVA is less than \(\alpha\), then reject \(H_{0}\) and accept \(H_{A}\).
If you are not familiar with this material, we suggest that you review course materials from your basic statistics course.
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There are 5 main steps in hypothesis testing: State your research hypothesis as a null hypothesis and alternate hypothesis (H o) and (H a or H 1 ). Collect data in a way designed to test the hypothesis. Perform an appropriate statistical test. Decide whether to reject or fail to reject your null hypothesis. Present the findings in your results ...
Learn what hypothesis testing is, how to calculate it, and why it is important for data analysis. See examples of null and alternative hypotheses, z-score, and p-value in this tutorial.
5.2 - Writing Hypotheses. The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ( H 0) and an alternative hypothesis ( H a ). When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the ...
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 ...
Learn what a statistical hypothesis is and how to test it using different methods. Find out the difference between null and alternative hypotheses, one-tailed and two-tailed tests, and Type I and Type II errors.
HYPOTHESIS TESTING. A clinical trial begins with an assumption or belief, and then proceeds to either prove or disprove this assumption. In statistical terms, this belief or assumption is known as a hypothesis. Counterintuitively, what the researcher believes in (or is trying to prove) is called the "alternate" hypothesis, and the opposite ...
Learn how to formulate null and alternative hypotheses for different statistical tests. The null hypothesis is the claim that there's no effect in the population, while the alternative hypothesis is the claim that there's an effect.
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.
Learn the general idea and basic procedures of hypothesis testing in statistics, with examples and analogies. Compare the critical value and P-value approaches to making decisions based on evidence.
Learn why and how to use hypothesis testing to make inferences about a population using a sample. Understand the basic terms, such as null and alternative hypotheses, p-values, significance level, and test statistic.
Test Statistic: z = x¯¯¯ −μo σ/ n−−√ z = x ¯ − μ o σ / n since it is calculated as part of the testing of the hypothesis. Definition 7.1.4 7.1. 4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true.
Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of a population. A hypothesis is a claim or statement about a characteristic of a population of interest to us. A hypothesis test is a way for us to use our sample statistics to test a specific claim.
Below these are summarized into six such steps to conducting a test of a hypothesis. Set up the hypotheses and check conditions: Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as H 0, which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is ...
Learn how to check if a claim about a population parameter is true or not using hypothesis testing. Find out the meaning of null and alternative hypotheses, significance level, test statistic, critical value and p-value, and the steps for testing proportions and means.
How do we test a claim or a hypothesis using statistical data? This webpage introduces the concept and procedure of hypothesis testing, a fundamental tool for inference in statistics. You will learn how to formulate null and alternative hypotheses, how to calculate test statistics and p-values, and how to interpret the results of hypothesis testing. This webpage is part of the Statistics ...
What does a statistical test do? Statistical tests work by calculating a test statistic - a number that describes how much the relationship between variables in your test differs from the null hypothesis of no relationship.. It then calculates a p value (probability value). The p-value estimates how likely it is that you would see the difference described by the test statistic if the null ...
Introduction to Hypotheses Tests. Hypothesis testing is a statistical tool used to make decisions based on data. It involves making assumptions about a population parameter and testing its validity using a population sample. Hypothesis tests help us draw conclusions and make informed decisions in various fields like business, research, and science.
Hypothesis testing involves two statistical hypotheses. The first is the null hypothesis (H 0) as described above.For each H 0, there is an alternative hypothesis (H a) that will be favored if the null hypothesis is found to be statistically not viable.The H a can be either nondirectional or directional, as dictated by the research hypothesis. For example, if a researcher only believes the new ...
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.
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.
Hypothesis testing. Hypothesis testing is a formal process of statistical analysis using inferential statistics. The goal of hypothesis testing is to compare populations or assess relationships between variables using samples. Hypotheses, or predictions, are tested using statistical tests. Statistical tests also estimate sampling errors so that ...
Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used ...
Step 7: Based on steps 5 and 6, draw a conclusion about H0. If the F\calculated F \calculated from the data is larger than the Fα F α, then you are in the rejection region and you can reject the null hypothesis with (1 − α) ( 1 − α) level of confidence. Note that modern statistical software condenses steps 6 and 7 by providing a p p -value.