This lecture explains how to conduct hypothesis tests about the mean of a normal distribution.
We tackle two different cases:
when we know the variance of the distribution, then we use a z-statistic to conduct the test;
when the variance is unknown, then we use the t-statistic.
In each case we derive the power and the size of the test.
We conclude with two solved exercises on size and power.
Table of contents
The assumptions are the same we made in the lecture on confidence intervals for the mean.
The sample is made of independent draws from a normal distribution.
Specifically, we observe the realizations of independent random variables , ..., , all having a normal distribution with:
unknown mean ;
known variance .
We test the null hypothesis that the mean is equal to a specific value :
To construct a test statistic, we use the sample mean
The test statistic, called z-statistic, is
A test of hypothesis based on it is called z-test.
We prove below that has a normal distribution with zero mean and unit variance.
The critical region iswhere .
Thus, the critical values of the test are and . How they are chosen is explained below.
The null hypothesis is rejected if , that is, if
Otherwise, it is not rejected.
The power function of the test iswhere is a standard normal random variable.
The notation indicates that the probability of rejecting the null is computed under the alternative hypothesis that the true mean is equal to .
The power function can be written aswhere we have definedAs demonstrated in the lecture entitled Point estimation of the mean, the sample mean has a normal distribution with mean and variance , given the assumptions we made above. If we de-mean a normal random variable and we divide it by the square root of its variance, the resulting variable ( in this case) has a standard normal distribution.
When evaluated at the point , the power function is equal to the probability of committing a Type I error, that is, of rejecting the null hypothesis when it is true.
This probability, called the size of the test, is equal to where the test statistic is a standard normal random variable.
Substite with in the power function and note that when .
As usual, the critical value is chosen so as to achieve a desired size .
In other words, we fix the size and then we find the critical value that solves
We explain how to do this in the page on critical values.
This case is similar to the previous one. The only difference is that we now relax the assumption that the variance of the distribution is known.
We observe the realizations of independent random variables , ..., , all having a normal distribution with
unknown mean ;
unknown variance .
We test the null hypothesis that the mean is equal to a specific value :
We construct the test statistic by using the sample meanand the adjusted sample variance
The test statistics, called t-statistic, is
The test of hypothesis based on it is called t-test.
We prove below that has a standard Student's t distribution with degrees of freedom.
The critical region iswhere .
Thus, the critical values of the test are and . How they are chosen is explained below.
We reject the null hypothesis if
Otherwise, we do not reject it.
The power function of the test is
where:
the notation indicates that the probability of rejecting the null is computed under the alternative hypothesis that the true mean is equal to ;
is a non-central standard Student's t distribution with degrees of freedom and non-centrality parameter equal to
The power function can be written aswhere we have definedGiven the assumptions made above, the sample mean has a normal distribution with mean and variance (see Point estimation of the mean), so that the random variablehas a standard normal distribution. Furthermore, the adjusted sample variance has a Gamma distribution with parameters and (see Point estimation of the variance). It follows thathas a Gamma distribution with parameters and . If we add a constant to a standard normal distribution and we divide the sum thus obtained by the square root of a Gamma variable with parameters and , we obtain a non-central standard Student's t distribution with degrees of freedom and non-centrality parameter . The variable has exactly this distribution, with parameter
The size of the test is equal to where the test statistic has a standard Student's t distribution with degrees of freedom.
The size of the test is obtained by evaluating the power function at the point :The statistic has a non-central standard Student's t distribution with degrees of freedom and non-centrality parameter equal toWhen , the non-centrality parameter is equal to and has a standard Student's t distribution.
As before, we choose the critical value so as to achieve a desired size .
Therefore, the critical value needs to solve the equation
The page on critical values explains how this equation is solved.
Below you can find some exercises with explained solutions.
Denote by the distribution function of a non-central standard Student's t distribution with degrees of freedom and non-centrality parameter equal to .
Suppose that a statistician observes 100 independent realizations of a normal random variable.
The mean and the variance of the random variable, which the statistician does not know, are equal to 1 and 4 respectively.
Find the probability that the statistician will reject the null hypothesis that the mean is equal to zero if:
she runs a t-test based on the 100 observed realizations;
she sets as the critical value.
Express the probability in terms of .
The probability of rejecting the null hypothesis is obtained by evaluating the power function of the test at :The notation indicates that the probability of rejecting the null is computed under the alternative hypothesis that the true mean is equal to . The variable has a non-central standard Student's t distribution with degrees of freedom and non-centrality parameterThus, the probability of rejecting the null hypothesis is equal to
Denote by the distribution function of a standard Student's t distribution with degrees of freedom, and by its inverse.
A statistician observes 100 independent realizations of a normal random variable.
She performs a t-test of the null hypothesis that the mean of the variable is equal to zero.
What critical value should she use in order to incur into a Type I error with 10% probability? Express it in terms of .
A Type I error is committed when the null hypothesis is true, but it is rejected. The probability of rejecting the null hypothesis is where is the critical value, and is a standard Student's t distribution with degrees of freedom. This probability can be expressed aswhere: in step we have used the fact that the density of a standard Student's t distribution is symmetric around zero. Thus, we need to set in such a way thatThis is accomplished by
Please cite as:
Taboga, Marco (2021). "Hypothesis tests about the mean", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-statistics/hypothesis-testing-mean.
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