Null and Alternative Hypothesis

Generally to understand some characteristic of the general population we take a random sample and study the corresponding property of the sample. We then determine whether any conclusions we reach about the sample are representative of the population.

This is done by choosing an estimator function for the characteristic (of the population) we want to study and then applying this function to the sample to obtain an estimate. By using the appropriate statistical test we then determine whether this estimate is based solely on chance.

The hypothesis that the estimate is based solely on chance is called the null hypothesis. Thus, the null hypothesis is true if the observed data (in the sample) do not differ from what would be expected on the basis of chance alone. The complement of the null hypothesis is called the alternative hypothesis.

The null hypothesis is typically abbreviated as H0 and the alternative hypothesis as H1. Since the two are complementary (i.e. H0 is true if and only if H1 is false), it is sufficient to define the null hypothesis.

Since our sample usually only contains a subset of the data in the population, we cannot be absolutely certain as to whether the null hypothesis is true or not. We can merely gather information (via statistical tests) to determine whether it is likely or not. We therefore speak about rejecting or not rejecting (aka retaining) the null hypothesis on the basis of some test, but not of accepting the null hypothesis or the alternative hypothesis. Often in an experiment we are actually testing the validity of the alternative hypothesis by testing whether to reject the null hypothesis.

When performing such tests, there is some chance that we will reach the wrong conclusion. There are two types of errors:

  • Type I – H0 is rejected even though it is true (false positive)
  • Type II – H0 is not rejected even though it is false (false negative)

The acceptable level of a Type I error is designated by alpha (α), while the acceptable level of a Type II error is designated beta (β).

We use the following terminology:

Significance level is the acceptable level of type I error, denoted α. Typically, a significance level of α = .05 is used (although sometimes other levels such as α = .01 may be employed). This means that we are willing to tolerate up to 5% of type I errors, i.e. we are willing to accept the fact that in 1 out of every 20 samples we reject the null hypothesis even though it is true.

P-value (the probability value) is the value p of the statistic used to test the null hypothesis. If  p < α then we reject the null hypothesis.

Critical region is the part of the sample space that corresponds to the rejection of the null hypothesis, i.e. the set of possible values of the test statistic which are better explained by the alternative hypothesis. The significance level is the probability that the test statistic will fall within the critical region when the null hypothesis is assumed.

Usually the critical region is depicted as a region under a curve for continuous distributions (or a portion of a bar chart for discrete distributions).

The typical approach for testing a null hypothesis is to select a statistic based on a sample of fixed size, calculate the value of the statistic for the sample and then reject the null hypothesis if and only if the statistic falls in the critical region.

One-tailed hypothesis testing specifies a direction of the statistical test. For example to test whether cloud seeding increases the average annual rainfall in an area which usually has an average annual rainfall of 20 cm, we define the null and alternative hypotheses as follows, where μ represents the average rainfall after cloud seeding.

H0: µ ≤ 20 (i.e. average rainfall does not increase after cloud seeding)

H1: µ > 20 (i.e. average rainfall increases after cloud seeding

Here the experimenters are quite sure that the cloud seeding will not significantly reduce rainfall, and so a one-tailed test is used where the critical region is as in the shaded area in Figure 1. The null hypothesis is rejected only if the test statistic falls in the critical region, i.e. the test statistic has a value larger than the critical value.

Right tailed significance test

Figure 1 – Critical region is the right tail

The critical value here is the right (or upper) tail. It is quite possible to have one sided tests where the critical value is the left (or lower) tail. For example, suppose the cloud seeding is expected to decrease rainfall. Then the null hypothesis could be as follows:

H0: µ ≥ 20 (i.e. average rainfall does not decrease after cloud seeding)

H1: µ < 20 (i.e. average rain decreases after cloud seeding)

Left tailed significance testing

Figure 2 – Critical region is the left tail

Two-tailed hypothesis testing doesn’t specify a direction of the test. For the cloud seeding example, it is more common to use a two-tailed test. Here the null and alternative hypotheses are as follows.

H0µ = 20

H1µ ≠ 20

The reasons for using a two-tailed test is that even though the experimenters expect cloud seeding to increase rainfall, it is possible that the reverse occurs and, in fact, a significant decrease in rainfall results. To take care of this possibility, a two tailed test is used with the critical region consisting of both the upper and lower tails.

Two tailed hypothesis testing

Figure 3 – Two-tailed hypothesis testing

In this case we reject the null hypothesis if the test statistic falls in either side of the critical region. To achieve a significance level of α, the critical region in each tail must have size α/2.

Statistical power is 1 – β. Thus power is the probability that you find an effect when one exists, i.e. the probability of correctly rejecting a false null hypothesis. While a significance level for type I error of α = .05 is typically used, generally the target for β is .20 or .10, and so .80 or .90 is used as the target value for power.

The general procedure for null hypothesis testing is as follows:

  • State the null and alternative hypotheses
  • Specify α and the sample size
  • Select an appropriate statistical test
  • Collect data (note that the previous steps should be done prior to collecting data)
  • Compute the test statistic based on the sample data
  • Determine the p-value associated with the statistic
  • Decide whether to reject the null hypothesis by comparing the p-value to α (i.e. reject the null hypothesis if p < α)
  • Report your results, including effect sizes (as described in Effect Size)

Observation: Suppose we perform a statistical test of the null hypothesis with α = .05 and obtain a p-value of p = .04, thereby rejecting the null hypothesis. This does not mean that there is a 4% probability of the null hypothesis being true, i.e. P(H0) =.04. What we have shown instead is that assuming the null hypothesis is true, the conditional probability that the sample data exhibits the obtained test statistic is 0.04; i.e. P(D|H0) =.04 where D = the event that the sample data exhibits the observed test statistic.

160 Responses to Null and Alternative Hypothesis

  1. Fernando Duarte Molina says:

    Mr. Charles Zaiontz,

    Thank you very much for sharing all your knowlegde with us, the lucky ones that have now a great source of information, excel tools, and motivation to understand statistics.
    Best regards.

    Fernando Duarte
    Santander, Colombia

  2. mohamed says:

    Hello Dear Charles Zaiontz,
    From the depth of my heart, thank you so much for sharing your knowledge with us. Your help was like a candle which lights our darkness statistic path. I really need your help how I can use T-Test for likert scale with 5 for two groups male and female. I do not need any formula, just what I have to do to run my excel Data Analysis to do a T-test


    Oakland, CA

  3. Aika says:

    Dear Mr. Zaiontz,

    I found myself in a very difficult situation, I have to make a report for my statistics coursework but my sole problem is commenting on my calculations (done in excel) by explaining the steps I have taken. Would you have a written sample of a statistics report that could guide me ? I have done all of my calculations already. The thing is I have word limitation on my report and I have difficulties with highlighting the most important steps when dealing with regression models or hypothesis testing in the report. From your professional experience, how would you explain your choice of significance level when it is not given to us?
    I would truly appreciate your help.

    Best regards,
    Aika, London

    • Charles says:

      For research reports you could follow the guidelines of the American Psychological Association (APA) (see e.g. You can find most of the details for free on the Internet. I don’t know of any guidelines for explaining the details of the calculations used since usually research reports don’t want to see all these details, but only the key results (as described in the APA guidelines).

  4. dee says:

    Hi. I’m doins a research on motivation and de-motivation among students. I’m using 3 independent variables. I’m testing how the 3 IV affect motivation and de-motivation. Should I use null hypotheses to test motivation and de-motivation? At the same time I’m using 2 dependent variables(motivation and de-motivation).
    Would appreciate your help. tq

    • Charles says:

      Sorry, but I need more information about what you are trying to accomplish before I am able to answer your question.

  5. João says:

    Thanks a lot for this site! I’m a psychologist from Brazil and I’m doing a study here that requires statistical analisys, and you’re helping me alot!

    Thanks again! Your site is the best one for learning how to use excel for statistical work!

  6. Gilles says:

    Dear Charles,

    at the end of the article, is seems that a small error is still present :
    “The general procedure for null hypothesis testing is as follows:
    State the null and alternative hypotheses
    Specify and the sample size”

    Maybe the “and” should be removed ?

    Thanks again for you wonderful website (and for your yesterday’s answer),


    • Charles says:

      Dear Gilles,
      It should say “Specify alpha and the sample size”. I have now made this correction on the referenced webpage. Thanks for catching this typing omission. I am pleased that you are enjoying the website and making valuable contributions as well.

  7. May I get some assistance on a QMB class by using Microsoft Excel.
    Best regards,
    Anorve Belizaire

  8. elham says:

    i have a critical question concerning above issue. can we say that we have two constructs of hypothesis namely , alternative,null hypothesis? the former is more abstract than null. i mean that the researcher should firstly maintain or reject null to touch abstract construct that’s to say,alternative research.put differently,alternative hypothesis is grounded on null- hypothesis. am i right?.i look forward to hearing from you

    • Charles says:

      I am not sure that I follow you completely, but I think that in some sense you are correct. Usually (although not always) you posit the null hypothesis with the goal of proving the complement, namely the alternative hypothesis. The alternative hypothesis is completely determined by the null hypothesis, namely the alternative hypothesis is true if and only if the null hypothesis is false. If you look at some older statistics text books, this wasn’t exactly the way it was stated, and may have been more aligned with your approach. In any case, today they are simply complements of each other and so if you think of the null hypothesis as concrete then the alternative hypothesis is just as concrete.

  9. Machelle says:

    I have this problem in a class and am in desire need of help. Thanks

    Share with your peers the null and alternative hypotheses for a decision that is relevant to your life. This can be a personal item or something at work. Additionally, identify the Type I and Type II errors that could occur with your decision‐making process.

    • Charles says:

      It is not possible for me to create hypotheses for your life. Just come up with something similar to the various null/alternative hypotheses that you see throughout the website.

  10. Vicky says:

    Hello Sir,
    i wanted to know that how is “non central distribution” used in the testing of hypothesis?

  11. Pamela says:

    Please I need assistance in my research work in data analysis and presentations. I have 3 objective as thus: what are the problems teachers are facing in teaching English in secondary school. What are the problems learners face in learning English. What are the solutions to the problems of teaching and learning English language in sec school. Then my hypothesis is there is no significant difference between the problems identified as affecting the teaching of English language and that of learning it in sec school in PH LGA. Please am using frequency count and percentage methods of data analysis. I need more explanation on it and how to calculate the hypothesis.

    • Charles says:

      I am sorry, but the information you have provided is not specific enough for me to offer any advice.

  12. Ori says:


    I was wondering if you could explain about the difference between a T Test with equal and unequal variances?

    I have two samples of visitors who visited two versions of our website over the last 3 months and their average order value.
    We have sampled 1805 visitors for the first version of our website and 614 visitors for our the second version.
    The two samples are independent (a visitor will always be redirect to the version he was assigned to)
    The samples’ variances are not equal.

    Can I assume that the population variances are not equal and therefore use the T Test for unequal variances?


    • Charles says:

      Especially with samples that are quite unequal in size, it is safe to use the t test with unequal variances. Even if the variances are pretty close to equal the two tests will yield quite similar p-values.

      • Ori says:

        Hi Charles,

        Which test provides more conservative results?
        Is there a guideline to when to use a test with unequal variances instead a test with equal variances?


        • Charles says:


          As you can see from Figure 4 of t test with unequal variances, if the two samples are equal in size the df for the t-test with unequal variance is always lower than the t-test with equal variance and the p-value is always higher.

          As I said previously, unless the variances are really different the test results will be almost the same. In general I tend to always use the unequal variance test for this reason. I don’t know of any specific guidelines.


  13. Joy says:

    Hi Charles,
    Please I need assistance to answer this question.
    In the test of difference and relationship, explain how to reject a null hypothesis and as well as accepting an alternative hypothesis.

    • Charles says:

      This is explained on the reference webpage. In general you set up a statistical test and calculate a p-value. If the p-value < alpha then you reject the null hypothesis. If something on the webpage is not clear to you, please ask a more detailed question about that concept. Charles

  14. Alec says:

    Charles, so the question I’m trying to answer is:
    When running a regression in Excel what do you use to determine if the null hypothesis is accepted or rejected? What parts of the regression output are used in the regression equation?

    While your description above definitely aids, I’m wondering if you had a more specified answer to my problem?


    • Charles says:


      Q. When running a regression in Excel what do you use to determine if the null hypothesis is accepted or rejected?
      A. Significance F

      Q. What parts of the regression output are used in the regression equation?
      A. The coefficients


  15. Cletus says:

    please, I need your assistance to answer this question. State ten null hypotheses and ten alternative hypotheses for me.

  16. Cynthia Jones says:

    This is very helpful but I’m still not sure how to calculate the power of my equation and how to calculate the probability of a Type II error.

  17. Haylee says:

    Given the following information… how do you extract the null hypothesis equation? Thanks

    A study was designed to compare three different methods for reducing stress. Sixty employees in a corporation were randomly assigned to each of the following groups for a two-week period: exercise, relaxation training, and management skills training (learning to set priorities, communicate effectively, resolve conflicts). Data consisted of subject scores on a measure of stress in the workplace taken after each two-week training period.

    • Charles says:

      It really depends on what you want to demonstrate. From your description, you probably want to test whether the various stress-reduction programs are significantly different in their ability to reduce stress. Usually this sort of analysis is done using ANOVA. The null hypothesis is that all the programs are equal in their ability to reduce stress.

  18. Madison says:

    Hi there,
    I having a lot of trouble with determining the p value. How do i find it with a sample size of 124 and a alpha level of 0.5. Also what test would i use to test the null hypothesis with the two nominated variables of stress levels and perceived stress levels (have to see if they are related). Also how do i state the assumptions for the test?
    Thanks kindly,

    • Madison says:

      Sorry also, step three above, how do i know which test to run?

    • Charles says:


      The referenced webpage is only intended to explain the general approach to hypothesis testing. The actual approach used depends on the specific hypothesis that you are trying to test. Much of the rest of the website addresses the various types of hypothesis testing that you might need to do.

      Regarding your specific problem, it sounds like you are trying to see whether there is a correlation between perceived and measured stress. This is likely to be addressed using a t test (or an equivalent correlation test). See the following webpages for more details:
      Paired Samples t Test
      One Sample Hypothesis Testing for Correlation.

      Also, it unlikely that you want an alpha of .5. You probably mean .05.


  19. Bryan Lee says:

    I have two samples from a batch of paint. One sample is taken from the manufacturer and another sample is taken from the same batch when sent to the applicator’s yard. The testing agency is the same for each test. What I am trying to show is that their is no significant difference in testing results taken from these separate locations. Is a paired t test the appropriate approach?

    • Charles says:

      Since you can’t look at each of the sample elements as paired in any way, the paired t test doesn’t seem appropriate. I guess that I would use a two sample t test (provided the normality assumption is met), although strictly speaking the samples are not completely independent.

  20. Thomas says:

    Dear Charles,

    I am working on my final master’s paper and have to do some t-tests.
    The idea is to test whether certain groups of people, classified with a self-reported score, the Need for Cognition, that helps to identify people as those who like intellectual tasks and to think problems through and those who don’t are more likely to answer another set of questions consistently with the same answer or whether they change their answer depending on how the question is asked and presented.

    I am not sure how to formulate my null hypothesis here. I expect those with a high Need for Cognition score to answer questions more consistently and not to be affected by how the objectively equivalent messages are framed/presented.

    Can you help me with the hypothesis formulation and then the decision which t-test (one-sided vs. two-sided) to choose?

    Thank you very much in advance!

    Best regards,

    • Charles says:


      I would need to see more details to answer your question definitively, but I can think of two possible approaches.

      (1) Null hypothesis: The mean score for the two group are the same. When in double, use a two tailed test. When you are pretty sure that mean score for group 1 can’t be less than the mean score for group 2, then you could use a one-tailed test.

      (2) Don’t view this problem as one of hypothesis testing of means, but as a comparison of ratings and use Cohen’s kappa or something similar.

      Probably the approach you are looking for is approach (1).


      • Thomas says:

        Thank you Charles!

        I think the two-tailed test for an equal mean score is the correct thing here.
        I used a hypothesized mean difference of zero, which should be the right test, correct? This means, I test if the mean difference between those two groups (high NFC versus low NFC group) is equal to zero, and if it is close enough to zero , the test result would tell me there is no mean difference. This would mean, people answer equally consistent no matter to which group they belong?

        My two-tailed critical value is 1.97 and the t-Stat is -2.02. Somewhere I read, if the t-Stat (-2.02) is below the negative critical value (-1.97), which is the case in my study, I shall reject the null hypothesis. Is this correct, too?
        In my case, then, the hypothesized mean difference of zero should be rejected, i.e. the two groups actually have a mean difference of larger than zero, hence the two groups performed differently consistently.

        Thanks again!


  21. ST says:

    Hi Thomas,
    I need some guidance here.
    I’m using f-test to check for the equality of variance between two sample of mean score of multifactor leadership questionnaire before deciding which t-test to perform. The p-value of each nine dimensions gave me mix of significant where three out of nine are greater than 0.05 and the rest are lower than 0.05. In this situation, can I conclude that the variance is not equal since majority fall within less than .05?
    Thanks for your help.

    • ST says:

      Hi Charles,
      Sorry, I have mistakenly address you wrongly.
      Please accept my apologies..

    • Charles says:

      I don’t completely understand the scenario, but it is very unlikely that it would be correct to conclude that the variances are not equal simply because the majority are less than .05.

      • ST says:

        Thanks Charles, I’m collecting data for two random samples using the questionnaire I mentioned earlier. In this questionnaire, there are nine dimensions each representing different leadership style. I’ve applied the f-test on each dimension to check for the equality of variances and below is what I got.

        Dimension F Significant
        IIA 1.9215 0.0392
        IIB 1.9215 0.0175
        IM 1.6443 0.0535
        IS 2.1763 0.0062
        IC 1.9863 0.0135
        CR 1.1428 0.0016
        MBEA 1.1428 0.3331
        MBEP 1.5012 0.0936
        LF 2.1530 0.0068

        From the significant value obtained, there are three dimensions having more than 0.05. That’s why I’m struggling to make the conclusion whether these two samples are having equal or unequal variances. Meanwhile, I have also computed the t-test two-sample equal and unequal variances, all dimensions except one (MBEA) are significantly below 0.05.
        I understand from your earlier feedback that it is likely that I could conclude the variances are not equal and I guess I’m not able to conclude to equal variances either. In this instances, do you have any other recommendation that I could use or test to analyse the two sample data? My objective of this study is to find out whether there is any differences between these two samples.
        Hope you could be in help. Thanks for your time.

        • Charles says:


          What I said previously is that even if the variances are equal the t test with unequal variances should be pretty accurate. Since the p-value > .05 for the t test for MBEA, then you cannot reject the hypothesis that the (population) means for MBEA are equal. Since the p-values for the other tests are less than .05, you conclude with 95% confidence that the means for these dimensions are equal.

          The only problem with this approach is that 9 tests, and so you have introduced an experiment-wise error of more than 5%. You might want to use a corrected alpha value as described on the webpage


  22. Aki says:

    Your notes were quite helpful to me but I have a worry,what in a case where there was statistical significance with a chi-square test n for the same variable,no significance with a logistic regression model, what then should the conclusions on the hypothesis be based on?

    • Charles says:


      I would have to see the specific results to answer precisely, but, in general, it can happen that you run two different statistical tests and get different, even contradictory, results. Sometimes this is because some assumption has not been met in one of the tests. Sometimes it is because the results are similar but the conclusions are different (e.g. the p-value for one test is .051 and the p-value for the other is .049). Sometimes one test is more accurate (e.g. the t test when the assumptions are met vs. a non-parametric test such as Mann-Whitney).

      In any case, this is the nature of statistical testing: it is probabilistic in nature. When this happens, you should report both results and draw whatever conclusions make sense, even that no clear-cut conclusion can be made.


  23. Terence says:


    Did students prefer to skip class the day before spring break , or the day after spring break ?

    John has reviewed about 200 fellow students selected at random in the University. He found that there were 120 students prefer to skip class the day before spring break , and 80 students were chosen to skip classes a day after spring break. Fifty of these individuals also have chosen to skip two days of spring break.

    What is the hypothesis null & alternative? How to calculate the P-Value manually?

  24. jenny says:

    in hypothesis testing, I assume that the null hypothesis is true in the sample or the population?

  25. christine says:

    i still cant understand about the alternative hypothesis. help?? please.

    • Charles says:

      To give a silly example, if the null hypothesis is “all dogs are aggressive” then the alternative hypothesis is the opposite, namely “not all dogs are aggressive”. Often you expect the alternative hypothesis to be true. If you need more help, please be more specific about the sort of help you need.

  26. Stella says:

    Our teacher asked this question to us and I really need to know the answer so if you can be of help, please do.
    “What is the main characteristic of the null and alternative hypothesis?”
    It’s only 1 main characteristic.

    • Charles says:

      I am afraid that you will need to provide this answer yourself, but hopefully the referenced website helps you with the response.

      • Stella says:

        I see. Thank you. It did help but I did have a bit of a trouble picking the best characteristic to pick based on what you wrote in the article.

  27. Raza says:

    if there is alternate hypothesis then why we use null hypothesis in statistics? kindly explain briefly

    • Charles says:

      It is very common that you are trying to refute the null hypothesis in order to show that alternative hypothesis is true. Recall that the alternative hypothesis is true if and only if the null hypothesis is false.

      • Raza says:

        i see . .! but why we try to reject or accept null hypothesis in ordered to prove alternate hypothesis accepted or rejected , , why not we directly proved or disproved alternate hypothesis . . .? little bit confusing this , , kindly guide it

  28. Maryam says:

    Why we use null hypothesis in statistics instead of alternative hypnosis???

    • Charles says:

      I believe that it is for historical reasons, but in any case the null and alternative hypotheses are flip sides of the same coin.

  29. anum says:

    plz answer this question
    why we donot test alternate hypothesis in statistics

  30. Kyle says:

    If we perform a one tailed test and our directional prediction was incorrect, do we fail to reject the null?

    I often see:
    H0: u1 = u2
    Ha: u1 > u2

    So let’s say the test shows we were wrong, in fact u1 < u2. But u1 does not equal u2, so it's difficult for me to want to fail to reject the null. I just did a test like this and I want to reject the null although I was directionally wrong.

    Is it the case that even though the null was written as (=), in actuality, there is a tacit agreement that we really mean (= OR <)? Because I really do see this often–the null being written only as (=) and not (= or ). I am having trouble finding a consensus on whether or not to reject the null when you chose the wrong direction. Thanks.

    • Charles says:

      You only perform a one-tailed test if you are sure that one side of the test is impossible. For the example you gave, you are assuming that u1 < u2 can't happen. If you are not sure of this, then you should perform a two tailed test. Charles

  31. Alexander says:

    Hello Charles. Could you please give me directions how to solve such a question:

    Test the null hypothesis that apples stock beta equals 1 using a two-tailed test. Make sure that you state the null and alternative hypotheses, the degrees of freedom used and the critical value from the Student’s t distribution.

    Significance level: 5 %
    coefficient: 1.28505377591414
    Standard error: 0.0723764939179485

    do i need to use a formula for calculating the hypothesis?

  32. Simon says:

    Hi Charles,

    The terminology of type I /II errors seems a bit counter-intuitive. If a test result is wrong, then I would agree to call that a ‘false’ test result (i.e. wrong/error= false and accordingly, type I and type II are both called ‘false’). However, if the null hypothesis H0 is ‘rejected’ by the test, then I would call that a ‘negative’ test result (i.e. rejected = negative). Hence the type I error event that a true Ho is rejected, would be called a ‘false negative’

    What is the alternative logic of calling a Type I error a ‘false positive ?

    • Charles says:

      Hi Simon,
      Here, the word positive is used since the null hypothesis is actually True. The word false is used since the conclusion is wrong, namely to reject the null hypothesis.

  33. naqeebullah says:

    pleas explain to me this question my problem is this when I face to any question in hypotheses I can not Analyse how to make null and alternate hypotheses and about choosing the tailed, which tailed should I choose .pleas make me clear about this doubt .

    thank you

    • Charles says:

      Null/alternative hypothesis: The best thing to do is look at lots of examples and see how these hypotheses are defined. There are numerous examples given throughout the website. See, for example, the t test.

      Tails: Generally you choose the two tailed test. Only when you are pretty sure (usually on theoretical grounds) that one of the tails won’t occur should you consider using a one-tailed test.


  34. L.Amaral Afonso says:

    April 6, 23:30

    Hi Charles

    I had just read your introduction on NHST.
    Excellent, you touch the main points, clearly , straightforward.
    When we see so much trash in literature for example “accept the null”,
    “the null is never attained, therefore the tests are useless”, and so on,
    I am happy to read from you what is not seemly obvious to some scientists/professors
    in particular Psychologists and Biologist as you surely had found. Well done Charles,
    Though we deliberated to retain 95% (say) probability to the Null Hypothesis
    and a scarce 5% to the Alternative we are never apt to “accept” H0, not whithstand
    found in a no few statistical literature. Jargon? could be, but dangerous
    for beginners. On contrary In case the test falls outside ill-named
    “acceptance interval” we reject H0 because its evident unlikeliness.


  35. L.Amaral Afonso says:

    This time

    ___pvalue less than alpha________significant________H0 rejected
    ___pvalue greater than alpha_____not significant_____fail to reject H0


  36. L.Amaral Afonso says:

    Charles, please
    Would, you so kind to cut after “Well done Charles” changing (April 6, 23:30)
    confusing/incorrect text by the following:
    Though we deliberated to retain 95% (say) probability to the Null Hypothesis
    and a scarce 5% to the Alternative we are never apt to “accept” H0, not whithstand
    found in a no few statistical literature. Jargon? could be, but dangerous
    for beginners. On contrary In case the test falls outside ill-named
    “acceptance interval” we reject H0 because its evident unlikeliness.
    Thanks so much, Charles (you are a very special person, all we can state)
    By the way: I would be really gratful for your corrections, indeed.

  37. Luís says:


    “Acceptance Interval”, indeed?
    There are lots of people that uses this designation
    I do prefer to name it “No-rejection”
    What do you think about?


  38. Dave says:

    Dear Sir, I have read somewhere that one can only test NULL hypothesis and one cannot test ALTERNATIVE hypothesis ……..Is it true? Why so ?

    • Charles says:

      The tests are all set up to test the null hypothesis, but keep in mind that the alternative hypothesis is true if and only if the null hypothesis is false.
      Also when you calculate the the power of a test, in some sense you are testing the alternative hypothesis.

  39. Ghulam Ishaq says:

    Respected sir,
    Would you please like to share some real life examples of a (alpha) in terms of average?

    Ghulam Ishaq
    Student of Business Administration
    The University of Poonch.

  40. afiqah says:

    Hello Sir. Do you know the answer to this?

    ” If you had a free hand, explain how you would like to conduct an appropriate hypothesis test and explain why your method is better. ”

    • Charles says:

      This sounds like a homework assignment. I prefer not to answer such questions, although some ideas can be found on the referenced webpage.

  41. okunola olabisi says:

    please i need examples of null hypothesis and alternative hypothesis as many as possible

  42. Luis A. Afonso says:

    Finally, I think to grasp the NHST behaviour in simple terms.

    The Null Hypothesis, H0: p=p0 should be thought has an
    approximate condition The same thing for the Alternative H1: p=p1
    Note that the latter value p1 is called strictly in order to calculate the Type II error. In fact. it is absolutely absent at the test formula which is deduced supposing the Null true.

    Sir Arthur Conan Doyle: Once you eliminate the impossible whatever remains, no matter how improbable, must be the truth.

    I modify slightly
    Once I eliminate the unlike, what remains has good chances to be plausible.

    (impossible to go further).

  43. Abdullah says:

    Should we use alternative or null hypotheses with t-test analysis???

    • Charles says:

      The test is for the null hypothesis, although since the alternative hypothesis is the complement of the null hypothesis it can also be viewed as a test for the alternative hypothesis.

  44. Anette kleppe says:

    Is IT possible to test alternative hypothesis as 5*beta >25000 ?
    The point is to test if an increase of 5 in the beta variable is larger than 25000

  45. Astrid Simone says:

    How do you state a null hypothesis saying the effect of beta will be greater than 100.000?

  46. Mel says:


    I have a question and hope you could help me. I have a null hypothesis as ‘there is a negative effect of X on Y’ and an alternative hypothesis as ‘there is a positive effect of X on Y’. P-value is set at .05.

    I use linear regression for the test and find that the unstandardized coefficient between X and Y is -.503 and Sig is .828. In this case, I should reject or accept the null hypothesis and why?

    Thanks for you help.

    • Charles says:

      Since sig = .828 > .05 = alpha, you would retain the null hypothesis. Note that you don’t “accept” the null hypothesis, but merely state that you don’t have sufficient reasons to reject it. This null hypothesis is that the corresponding regression coefficient is zero.
      Whether this null hypothesis is the same as your null hypothesis is not completely clear to me without further information.

  47. Jocelyn says:

    I have trouble finding the critical value and the TV in every problem. I’m hoping you could help me…
    For example, One hundred people were placed on a special exercise program. After one month, 61 lost weight, 11 gained and 28 weighed the same as before. Test the hypothesis that the exercise program is effective at alpha=0.02 (Note: It will be effective of fewer than 50% of the people did not lose weight.)

    • Charles says:

      For this problem, you need to decide which test to use to test the null hypothesis that at least 50% of the people lost weight. Hint: Possible choices are the binomial test or one-sample t test.

  48. Ashwini S says:

    Hello! Charles
    My final year project is ‘A comparative study on direct investment in equity and money market mutual fund’. 1)I am finding difficulty in framing hypothesis, could you please help me out & which of analytical tools to be used (apart from mean, standard deviation, beta n sharp index)
    2)Ho- investment in money market mutual fund is more significant than direct investment in equity.
    Ha- investment in money market mutual fund is not significant than direct investment in equity ( is this correct)
    Note: selected top 5 companies equity & mutual funds daily script value & NAV for 5 years are taken into consideration.

    • Charles says:

      I don’t quite understand your statement of the null hypothesis. Do you mean the following?
      Ho: investment in money market mutual fund has a higher return than direct investment in equity

  49. Miguel says:

    Dear Charles,

    As I understand it, the null hypothesis (H0) should never be stated with an inequality sign (), but always with an equal sign (=), due to the fact that there needs to exist a defined parameter to be tested for likelihood.

    In the case of the one-tailed hypotheses, you wrote:
    H0: µ ≥ 20 (i.e. average rainfall does not decrease after cloud seeding)
    H1: µ 0

    In which R(a) is one correlation coefficient and R(b) another one, of the variables a and b against a common variable c. Now, I am able to reject the null hypothesis in almost every case, but there are some cases where R(a)-R(b)=0 (so H0 is not rejected), but also some other cases where R(a)-R(b)<0. Now, my question is:

    Are both results [R(a)-R(b)=0 and R(a)-R(b)<0] equal motives for H0 rejection? In other words, should I treat R(a)-R(b)<0 as a rejection of H0, or differently?

    Best regards,
    Miguel Serra
    Porto, Portugal

  50. Miguel says:

    Dear Charles,

    I’m sorry, parts of my previous messages didn’t appear on the post (too much text?)

    My hypothesis is:
    H0: R(a)-R(b) = 0
    H1: R(a)-R(b) > 0

    Perhaps with this aditional info we can understand my question. Otherwise I’ll post it again.

    Many thanks,

  51. Miguel says:

    I’m sorry, the text was incomplete.

    Three variables: a, b, and c.

    H0: R(a,c)-R(b,c) = 0
    H1: R(a,c)-R(b,c) > 0

    My question is: I heard that you should not put inequality signs in H0, because you need to test a defined parameter. You should put inequality signs only in H1. But then, what to do with the cases where R(a,c)-R(b,c) < 0?
    These are naturally included in H0, right? Because H1 only admits positive values, hence everything else is part of H0. Is this correct?


  52. what is null hypothesis

    • Charles says:

      Sorry, but I don’t understand your question. The null hypothesis is whatever you want it to be as described on the referenced webpage.

  53. Anthony says:

    Suppose a chemical company has developed a catalyst that is meant to reduce reaction time in a chemical process. For a certain chemical process, reaction time is known to be 150 seconds. The researchers conducted an experiment with the catalyst 40 times and measured reaction time. The researchers reported that the catalysts reduced reaction time with a P-value of 0.02.

    • Anthony says:

      1) Identify the null and alternative hypotheses.
      2) Explain what this result means. Do you believe that the catalyst is effective? Justify and explain your reasoning.

      I need extreme help with this problem

      • Charles says:

        1) Probably the null hypothesis is that the catalyst does not reduce reaction time and the alternative hypothesis is that it does
        2) If alpha = .05, then p-value = .2, means that you have a significant result (which supports the alternative hypothesis that the catalyst is effective).

  54. Khanyi Mnisi says:

    Hi Charlse,
    I have a question, what is the null and alternative hypothesis from reading this study case:
    According to theory, different positions occupied by personnel in an organisation strongly inform their levels of
    job satisfaction. Job satisfaction refers to the persons feelings towards their work, and is intimately tied-up
    with their motivation towards getting their work done. It is defined as “the pleasurable emotional state
    resulting from the appraisal of one’s job as achieving or the achievement of one’s job values”. Job satisfaction
    is determined by factors such as job design, responsibility, level of difficulty, recognition, ability to progress,
    task identity and so forth. These factors are all part of the position held by an employee in an organisation.
    Considering the above, you are the HR manager at a company, and wish to determine if some employees
    occupying certain positions experience more job satisfaction than others. You administer a questionnaire to all
    employees at your company and develop an overall score measuring job satisfaction.
    You record the employee’s position as:
     General Worker
     Junior Management
     Middle Management
     Senior Management
    You develop an aggregate score for Job Satisfaction. You sample 50 employees

    • Charles says:

      You should be able to state the null and alternative hypotheses based on “considering the above, you are the HR manager at a company, and wish to determine if some employees occupying certain positions experience more job satisfaction than others”.

  55. Katrina says:

    My question is the original claim is that the mean amount of organic matter in soil is 3%. the competing claim is that the mean amount of organic matter in soil is something other than 3%.

    State the null hypothesis, H0, in symbolic form: I wrote u=0.03
    State the alternative hypothesis, H1, in symbolic form: I wrote u<0.03

    Am I right for both answers? *u is pronounce mu for sample mean*


  56. Kabeer says:

    Hi Charles,
    I have a query. I’m conducting a market research wherein i have 4 factors, say A,B,C and D affecting one thing, for example, say factors contributing to brand loyalty.
    Now, for A,B,C & D, i have 70 responses wherein the users have given a ‘1’ or a ‘0’, a 1 indicating that particular factor matters and 0 meaning it doesnt matter. (I have 70 rows of 4 columns of 1s and 0s in excel)
    Now, i want to find which factors( out of A,B,C,D) are relevant for brand loyalty using hypothesis testing.
    Which tests should i do? ( chi-square, 1 way anova or any other tool on excel?)
    My current hypothesis is : Factors A,B,C,D are significant for brand loyalty. Does it need to be changed?
    Please provide some insights.

    • Charles says:

      I don’t know of any absolute criteria for brand loyalty, and so I don’t know of any test which will determine which factors are significant for brand loyalty. You can test whether there are significant differences between the ratings of the four factors or correlations between the factors, but these don’t determine whether a particular factor is significant. You can, of course, set your own criteria (e.g. at least x% of the raters find a particular factor important), especially if you have earlier studies to support your criteria.

  57. Iqra says:

    Hi charles,
    I am conducting reseach on franchise (fast food) & selected two samples of customers, one sample of families and the second sample of singles ( I mean university age students). The sample size of each group is 10 and level of significance of 0.05. could you please help me for developing hypothesis for following data;
    Question Family singles/friends
    what they have ordered?
    pizza(regular) 3 2
    (small) 0 4
    (large) 7 4
    & could you please tell me method for identifying that our data is one- tailed or two tailed.

    • Charles says:

      The hypothesis depends on what you want to test. Only you can determine this. I am not able to look at your data and tell you what hypothesis you should use.

  58. Huma says:

    Dear Charles,

    Please interpret the following F-test. As P-value is greater than 0.05. Are the variances equal and I should accept my Null Hypothesis?

    data: org.batch1[, i] and org.batch2[, i]
    F = 1.1828, num df = 149, denom df = 134, p-value = 0.322
    Null Hypothesis: True ratio of variances for the two batches is equal
    alternative hypothesis: true ratio of variances is not equal to 1
    95 percent confidence interval:
    0.8479508 1.6450027
    sample estimates:
    ratio of variances

  59. Mofazzal Haider says:

    Sir ,
    I am HAIDER an MBA student and have to complete a research paper. My topic is ( condition of X football team). AS I know that hypothesis is an assumption(possible answer) of the research question and it has two variables dependent and independent and two hypothesis null(positive statement) and alternative (negative statement) hypothesis. Basically hypothesis create based on research question .
    My research question is ( Why x football team is week?).
    According to my assumption possible answer of the question would be ” x football team is week because of well coaching staff. Now i make hypothesis
    Now i do my hypothesis
    H1:The X football team will be strong if they hire good coaching staff.
    H0:The X football team will not be strong if they hire good coaching staff. /
    H0:The X football team will not be strong even if they hire good coaching staff.\
    Am i right or not. What could be?
    please help me.

    Best regards.

    Dhaka, Bangladesh.

    • Charles says:

      You list two null hypotheses (H0) and one alternative hypothesis (H1). Each null hypothesis should have one complementary alternative hypothesis.
      Here is a potential approach:
      H0: The strength of a football team is independent of the quality of its coaching staff
      H1: The strength of a football team depends on the quality of its coaching staff
      Essentially you are trying to support the alternative hypothesis by showing there is a significant difference between the strength of two football teams whose coaching staffs have different qualities (two tail version).
      You can rephrase to make a one tailed statement.
      Note that you will need to make the notions of strength and quality more precise (preferably quantifiable).

      • syed nasir shah says:

        dear sir,
        i am M.Phil student of bio statistics and epodemiology. yesterday i had discission with my snior. he sayr in both one tail and two tail situations the null is same mu= mu not
        i say this null is true for two tailed situatiuoi only. while for one tail , in addition to equality, one has to add opposite of alternate hypothesis as well. am i right or not, nasir shah

  60. Srini says:

    Dear Charles,
    I have a stupid question. I need to test H0: µ ≠ 0 against Ha: µ = 0 and get the p-value. How do I conduct such a test. Is it just 1-p-value when H0: µ = 0. And how can I specify that in Excel or Stata. Thanks

  61. Lisa says:

    If my mud alt hypothesis calls for a right tailed test is it possible for my t-score to be a negative number?

  62. Shailesh Parab says:

    Dear Charles,

    Sorry, If my question might sound stupid. With respect to the statement, “The hypothesis that the estimate is based solely on chance is called the null hypothesis.” What actually mean by “solely on chance”? Can you please elaborate.

    • Charles says:

      This is actually a pretty deep and philosophical question. Let me try responding with a simple example. If you toss a coin 1,000 times, you expect on average that 500 times heads will appear and 500 times tails. In reality an exact 500-500 split almost never happens. A split of 509-491 is probably due to chance, i.e. a random occurrence. If the split is say 809-191, this is very unlikely to be simply due to chance. The probability of such an event occurring is so low, that it make sense to suspect that the coin is biased.

  63. Barbara Lawson says:

    Hi Charles,

    I have read all the help that you have given others and I am doing my final paper project that is due this coming Sunday and I am having a hard time with this scenario in finding the null and alternative of this scenario and I have to draw and produce a Scatterplot of the data. Here is the scenario:
    What’s makes a person a leader? A social psychologist hypothesizes that the extent to which people perceived to be leaders is positively related to the amount of talking they do, regardless of what they are actually saying. She conducts an experiment in which groups of participants solve a problem together. One of the participants from each group is a confederate who speaks a different number of sentences for different groups, half of which are designed to help the groups to solve the problems and the other half are not helpful. Following the group work, the confederate is rated by the other participants on his and her leadership ability (1=low, 10=high).

    What I came up with for the null hypothesis If the number of words spoken makes a person a leader (alternative) then the words that are spoken effects leadership ability.
    I’m not good at this because I get mixed up with dependent and independent variable when it comes to applying my data.
    In the data that I am working with you have eight groups, with the number of sentences spoken and rating of leadership here are the following data that represents the findings of this fictional research.
    Number of sentences spoken
    1- 4
    2- 10
    3- 27
    4- 22
    5- 5
    6- 6
    7- 18
    8- 15
    Rating of Leadership
    1- 5
    2- 6
    3- 7
    4- 9
    5- 2
    6- 3
    7- 8
    8- 4
    Whatever help that you can provide for me is greatly appreciated and may you have a blessed day, thank you in advance,

    Barbara Lawso

    • Charles says:

      It is not clear to me what hypothesis this test is addressing. Usually, you state your hypotheses first and then you construct the experiments necessary to gather information to support or refute the hypotheses.
      It seems like you are trying to determine whether there is a difference in the ratings of the leaders based on the number of statements they make (or perhaps based on the percentage of erroneous statements). If so the null hypothesis is that there is no difference.

  64. Mary MacPherson says:

    I am an IB Psychology teacher and my students generated a directional hypothesis based on a study they were attempting to replicate. Their results were significant but in the inverse direction of the alternative hypothesis. What is the appropriate way to deal with this.

    • Charles says:

      I don’t quite know what you mean by the”results were significant but in the inverse direction of the alternative hypothesis.”
      I also don’t know what you mean “deal with this”. Are you asking (1) how to interpret such a result, or (2) what to do about it, or (3) something else?

  65. Alex Cartagena says:


    Can I create a T-Test with this null hypothesis, “who is less likely to stop properly at a stop sign, men or women? Or men are less likely to stop properly at a stop sign than women. What is the next step? I thank you in advance for your help.

  66. Emma Ogbogu says:

    How do you approach a situation like this:
    a. Calculate daily returns of the DJIA index and the downloaded stocks over the period under study (I have this info on a spread sheet already)
    b. For each of the selected stocks and the index perform a Student’s T test, calculate the p-value and t-value and test the Null Hypothesis that the mean daily stock return is zero

    for the a. I can handle but for the b. I don’t know how to approach it. Is it a paired test. independent or

  67. Heather says:

    I need help in writing a null and alternative hypotheses for a t-test. Could you help me?

    The question was investigated of whether pleasant aromas help a student learn better. All 21 participants learned both under a condition of smelling nothing and under a condition of smelling a floral scent. Counterbalancing was followed so that some participants learned without the scent first and some learned with the scent first. All participants were apprised that the scents were “safe” and that if they wished they could leave the study at any time. Data in terms of “time (in seconds) to complete a pencil and paper maze”shown below

  68. Shaneta Washington says:

    Hi Charles,

    I appreciate you sharing your knowledge on statistics, but I am having trouble with a few problems.
    Develop the null and alternative hypothesis for each of the following situations:
    – Mr. Rogers would like to know if there is sufficient evidence to conclude that more than 60% of stocks in the NYSE increase on days that the DJIA increases.
    – A company that manufactures one-half inch bolts selects a random sample of bolts to determine if the diameter of the bolts differs significantly from the required one-half inch.
    – A company that manufactures safety flares randomly selects 100 flares to determine if the flares last at least 3 hours on average.

    • Charles says:

      These problems are very similar to the cloud seeding / rainfall example on the referenced webpage.
      What do you think the answers to each of these three situations are?

  69. Ali Asadi says:

    Dear Dr. Zaiontz,
    Thank you for such valuable information you have shared.
    I have a question. I have conducted an experimental research with a null hypothesis, which was later rejected after I carried out a t-test. I am now wondering if at this stage it is possible to replace my null hypothesis with an alternative hypothesis, and then in the results section “accept or support” the alternative hypothesis? (A reviewer has asked me to do so).
    Thanks a lot.

    • Charles says:

      Theoretically you could do this, but the typical statistical test that you use usually won-t accept such a change in the null and alternative hypotheses roles.

  70. john says:

    can you help me solve this

    1. A DTI representative wants to test at 99% level of confidence whether the average content of soda is less than 330ml as indicated in the label.

    A. State the null and alternative hypothesis in words
    B. State the null and alternative hypothesis in symbol
    C. Probability of type error
    D. Conclusions where Ho is rejected
    E. Conclusions where Ho is not rejected

    2. Write the null or alternative hypothesis
    A. Hom = 10.2, what is Ha
    B. What is Ho if Ha= m not equal to 32.6
    C. What is Ho if Ha= m>45
    D. What is Ho if Ha is not equal to 0.87

    • Charles says:

      This looks like a homework assignment. What do you think the answers to the questions are? I can help you, but you need to put some effort into this too.

  71. Mike says:

    Hello Charles,

    Lets assume I have the following research hypotheses:

    Null Hypothesis: Men do not engage in gossips
    Alternative Hypothesis: Men engage in gossips.

    Assuming based some sample size, I find out that not all men engage in gossip but just a percentage of (or some) men engage in gossips (for eg, 20,30, 35 or 50). How do I conclude on the Hypotheses? Can I reject the Null Hypothesis because some men engage in gossip or is there any other test I have to do in order to arrive at a decision.

    • Charles says:

      The null hypothesis that you are using is a bit vague. When you say that men do not engage in gossip, do you mean zero? Obviously, in this case, even if one man in your sample gossiped this would refute your null hypothesis. If I replaced zero by some other value, say 1%, then I can perform a statistical test to determine whether there is a statistically significant difference between the percent of men (in the population) that gossip and 1% (two-tailed test). I could perform a statistical test to determine whether more than 1% of men (in the population) gossip with some level of confidence, say 95% (a one-tailed test).
      I guess I could also replace 1% by 0%, as long as I use this sort statistical interpretation (instead of the one where one counter-example is sufficient to refute the null hypothesis).
      In the situation described, I might use a proportion distribution (essentially a binomial distribution) to test my hypothesis. If say I have a sample of 100 men and 4 of them gossip, can I conclude that less than 1% of men in the general population gossip? My null hypothesis is that more than 1% gossip and I try to use my sample to gain evidence to refute the null hypothesis. In the test you assume the null hypothesis, namely that p = .01 is the probability that a man gossips and then use the binomial distribution to see how likely that in this case 4 out of 100 men in the sample gossip. Since =1-BINOM.DIST(4,100,0.01,TRUE) returns a value of .003 and that this value is much less than .05 (1-.05 is the usual 95% confidence level), we would conclude with 95% confidence that less than 1% of the men gossip. This is called a significant result.
      If in our sample 2 out of 100 men gossiped, then since 1-BINOM.DIST(2,100,0.01,TRUE) = .079 > .05, we can’t conclude with 95% confidence that fewer than 1% of the men gossip.

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