Jul 05, 2018 Another definition is “ (total variance explained by model) / total variance.” So if it is 100%, the two variables are perfectly correlated, i.e., with no variance at all. A low value would show a low level of correlation, meaning a regression model that is not valid, but not in all cases. What does land variance mean? A land variance involves using or developing the property in a way that deviates from the municipality's rules. If a property has a land variance the town granted the owner permission to use their land in a way that isn't normally permitted.

This lecture presents some examples of Hypothesis testing, focusing on tests of hypothesis about the variance, that is, on using a sample to perform tests of hypothesis about the variance of an unknown distribution.

Table of contents

Normal IID samples - Known mean

In this example we make the same assumptions we made in the example of set estimation of the variance entitled Normal IID samples - Known mean. The reader is strongly advised to read that example before reading this one.

The sample

The sample is made of independent draws from a normal distribution having known mean and unknown variance . Specifically, we observe realizations , ..., of independent random variables, ..., , all having a normal distribution with known mean and unknown variance . The sample is the -dimensional vector , which is a realization of the random vector.

The null hypothesis

We test the null hypothesis that the variance is equal to a specific value :

The alternative hypothesis

We assume that the parameter space is the set of strictly positive real numbers, i.e., . Therefore, the alternative hypothesis is

The test statistic

How To Calculate Variance

To construct a test statistic, we use the following point estimator of the variance:

The test statistic isThis test statistic is often called Chi-square statistic (also written as -statistic) and a test of hypothesis based on this statistic is called Chi-square test (also written as -test).

The critical region

Let and . We reject the null hypothesis if or if . In other words, the critical region isThus, the critical values of the test are and .

The power function

The power function of the test iswhere is a Chi-square random variable with degrees of freedom and the notation is used to indicate the fact that the probability of rejecting the null hypothesis is computed under the hypothesis that the true variance is equal to .

The power function can be written aswhere we have definedAs demonstrated in the lecture entitled Point estimation of the variance, the estimator has a Gamma distribution with parameters and , given the assumptions on the sample we made above. Multiplying a Gamma random variable with parameters and by one obtains a Chi-square random variable with degrees of freedom. Therefore, the variable has a Chi-square distribution with degrees of freedom.

The size of the test

When evaluated at the point , the power function is equal to the probability of committing a Type I error, i.e., the probability of rejecting the null hypothesis when the null hypothesis is true. This probability is called the size of the test and it is equal to where is a Chi-square random variable with degrees of freedom (this is trivially obtained by substituting with in the formula for the power function found above).

Normal IID samples - Unknown mean

This example is similar to the previous one. The only difference is that we now relax the assumption that the mean of the distribution is known.

The sample

In this example, the sample is made of independent draws from a normal distribution having unknown mean and unknown variance . Specifically, we observe realizations , ..., of independent random variables , ..., , all having a normal distribution with unknown mean and unknown variance . The sample is the -dimensional vector , which is a realization of the random vector .

The null hypothesis

We test the null hypothesis that the variance is equal to a specific value :

The alternative hypothesis

We assume that the parameter space is the set of strictly positive real numbers, i.e., . Therefore, the alternative hypothesis is

The test statistic

We construct a test statistic by using the sample meanand either the unadjusted sample varianceor the adjusted sample variance

The test statistic isThis test statistic is often called Chi-square statistic (also written as -statistic) and a test of hypothesis based on this statistic is called Chi-square test (also written as -test).

The critical region

Let and . We reject the null hypothesis if or if . In other words, the critical region isThus, the critical values of the test are and .

The power function

The power function of the test iswhere the notation is used to indicate the fact that the probability of rejecting the null hypothesis is computed under the hypothesis that the true variance is equal to and has a Chi-square distribution with degrees of freedom.

The power function can be written aswhere we have definedGiven the assumptions on the sample we made above, the unadjusted sample variance has a Gamma distribution with parameters and (see Point estimation of the variance), so that the random variablehas a Chi-square distribution with degrees of freedom.

The size of the test

The size of the test is equal to where has a Chi-square distribution with degrees of freedom (this is trivially obtained by substituting with in the formula for the power function found above).

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Denote by the distribution function of a Chi-square random variable with degrees of freedom. Suppose you observe independent realizations of a normal random variable. What is the probability, expressed in terms of , that you will commit a Type I error if you run a Chi-square test of the null hypothesis that the variance is equal to , based on the observed realizations, and choosing and as the critical values?

The probability of committing a Type I error is equal to the size of the test:where has a Chi-square distribution with degrees of freedom. ButThus,If you wish, you can utilize some statistical software to compute the values of the distribution function. For example, with the MATLAB commands chi2cdf(65,39) and chi2cdf(15,39) we obtainAs a consequence, the size of the test is

Low Variance Means

Exercise 2

Make the same assumptions of the previous exercise and denote by the inverse of . Change the critical value in such a way that the size of the test becomes exactly equal to .

Replace with in the formula for the size of the test:You need to set in such a way that . In other words, you need to solvewhich is equivalent toProvided the right-hand side of the equation is positive, this is solved byIf you wish, you can compute numerically. From the previous exercise we know thatTherefore, we need to computeIn MATLAB, this is done with the command chi2inv(0.0444,39), which gives as a result

Exercise 3

Make the same assumptions of Exercise 1 above. If the unadjusted sample variance is equal to 0.9, is the null hypothesis rejected?

In order to carry out the test, we need to compute the test statisticwhere is the sample size, is the value of the variance under the null hypothesis, and is the unadjusted sample variance.

Thus, the value of the test statistic isSince and , we have thatIn other words, the test statistic does not exceed the critical values of the test. As a consequence, the null hypothesis is not rejected.

How to cite

Please cite as:

Taboga, Marco (2017). 'Hypothesis tests about the variance', Lectures on probability theory and mathematical statistics, Third edition. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-statistics/hypothesis-testing-variance.