1.7. Tests of the variance of a normal distribution

In addition to the need for tests based on the sample mean and sample proportion, there are a number of situations where we want to determine if the population variance is a particular value or set of values. The basis for developing particular tests lies in the fact that the random variable

follows a Chi-square distribution with degrees of freedom.

The value of the test statistic is calculated as

We are given a random sample of n observations from a normally distributed population with variance . If we observe the sample variance , then the following tests have significance level :

1. To test either null hypothesis

or against the alternative

the decision rule is

Reject if

2. To test either null hypothesis

or against the alternative

the decision rule is

Reject if

3. To test the null hypothesis

against the two sided alternative

the decision rule is

Reject if or

where is a Chi-square random variable and .

Example:

Variance of yearly earnings of all state employees for all 40 states is

$49000 square dollars. A sample of 29 employees selected from state A produced a variance of their earnings equal to $600 000 square dollars. Test at 5% significance level if the variance of yearly earnings of state employees in state A is different from $490 000 square dollars. Assume that the yearly earnings of all state employees in state A have an (approximate) normal distribution.

Solution:

From the given information,

; ;

The null and alternative hypotheses are

We use Chi square distribution to use. The decision rule is

Reject if or

; ;

Then from Table 3 of appendix we obtain

and

The value of the test statistic is

The value of the test statistic 34.286 is between the two critical values of , 15.308 and 44.461, and falls in the nonrejection region. Consequently we fail to reject and conclude that the population variance of yearly earnings of all employees in state A is not different from 490000 square dollars.

Exercises

1. A sample of 24 observations selected from a normally distributed population produced a sample variance of 12.

a) Write the null hypothesis and alternative hypothesis, and decision rule to test if the population variance is different from 10.

b) Using , find the critical values of . Show the rejection and nonrejection regions on a Chi-square distribution curve.

c) Using the 5% significance level, will you reject the null hypothesis stated in part a)?

2. A sample of 25 observations selected from a normally distributed population produced a sample variance of 18. Using the 2.5% significance level, test hypothesis if the population variance is less than 25.

3. Usually people do not like waiting in line for service for a long time.

A bank management does not want the variance of the waiting time for her customers to be higher than 4.0 square minutes. A random sample of 25 customers taken from this bank gave the variance of the waiting times equal to 7.9 square minutes. Test at 1% significance level if the variance of the waiting time for all customers at this bank is higher than 4.0 square minutes. Assume that the waiting time for all customers is normally distributed.

4. Test against with in each case

a)

b) ;

5. A sample of seven observations taken from a population produced the following data

10; 8; 13; 15; 6; 8; 13

Assuming that the population from which this sample is selected is normally distributed, test at 2.5 significance level if the population variance is different from 10.

6. A drug manufacturer requires that the variance for a chemical contained in the bottles of certain type of drug should not exceed 0.03 square grams. A sample of 25 such bottles gave the variance for this chemical as 0.06 square grams. Test at the 1% significance level if the variance of this chemical in all such bottles exceeds 0.03 square grams. Assume that the amount of this chemical in all such bottles is (approximately) normally distributed. Find and interpret p-value of this test.

7. A random sample of ten students was asked, in hours, for time they spent studying in the week before final exams. The data are as follows:

28; 57; 42; 35; 61; 39; 55; 46; 49; 38

Assuming that the population distribution is normal, test at 5% significance level against two sided alternative the null hypothesis that the population standard deviation is 10 hours

8. Company claims that its employees earns a mean of at least $40 000 in a year and that the population standard is no more than $6 000. Earnings of a random sample of nine employees of this company produced

and

where are measured in thousands of dollars and population distribution can be assumed to be normal. Test at 10% significance level the null hypothesis that the population standard deviation is at most $6 000.

9. State whether each of the following statement is true or false

a) The significance level of a test is the probability that the null hypothesis is false.

b) A Type I error occurs when a true null hypothesis is rejected.

c) A null hypothesis is rejected at the 0.025 level, but is accepted at the 0.01 level. This means that p-value of the test lies between 0.01 and 0.025.

d) If a null hypothesis is rejected against an alternative at the 5% level, then using the same data, it must be rejected against that alternative at the 1% level.

f) If a null hypothesis is rejected against an alternative at the 1% level, then using the same data, it must be rejected against that alternative at the 5% level.

g) The p- value of a test is the probability that the null hypothesis is true.

Answers

1. b) reject if or ; b) ; do not reject ; 2. ; do not reject ;3. ; reject ; 4. a) ; reject ; b) ; accept ;

5. ; do not reject ; 6. ; reject ; reject for ; 7. ; accept ; 8. ; accept ;

9. a) false; b) true; c) false; d) false; f) true; g) false.