Hypothesis Tests for Two Population Standard Deviations Section 11.4

Objectives Find critical values of the F distribution Perform a hypothesis test for two population standard deviations

Find critical values of the F distribution Objective 1 Find critical values of the F distribution

Notation and Assumptions The tests we have studied so far have involved means or proportions. Occasionally it may be desirable to test a null hypothesis that two populations have equal standard deviations. A method is available in the special case where both populations are normal. We use the following notation: 𝜎 1 and 𝜎 2 denote the standard deviations of the two populations 𝑠 1 and 𝑠 2 denotes the sample standard deviations 𝑛 1 and 𝑛 2 denotes the sample sizes The method requires the following assumptions: Assumptions: We have independent random samples from two populations. Both populations are normal.

Test Statistic The null hypothesis for the test for two population standard deviations is 𝐻 0 : 𝜎 1 = 𝜎 2 . The alternate hypothesis will take one of the follow forms: 𝐻 1 : 𝜎 1 < 𝜎 2 , 𝐻 1 : 𝜎 1 > 𝜎 2 , or 𝐻 1 : 𝜎 1 ≠ 𝜎 2 The test statistic for this type of test is denoted by the letter F. The value of F is computed using the sample variances 𝑠 1 2 and 𝑠 2 2 . 𝐹= larger of 𝑠 1 2 and 𝑠 2 2 smaller of 𝑠 1 2 and 𝑠 2 2 When 𝐻 0 is true, 𝜎 1 = 𝜎 2 . Therefore the sample standard deviations 𝑠 1 and 𝑠 2 are on average approximately the same size, so the test statistic F is likely to be near 1. If the test statistic is much greater than 1, we will reject 𝐻 0 .

F distribution In order to use F, we must know its distribution when 𝐻 0 is true. This distribution is called an F distribution. Statistics that have an F distribution are ratios of quantities, such as the ratio of two variances. The F distribution has two values for the degrees of freedom: one associated with numerator and one associated with the denominator. The F distribution is skewed to the right.

Degrees of Freedom and Critical Value Since the F distribution has two degrees of freedom associated with it, a common manner to denote the values of the degrees of freedom is to use subscripts. The first value in the subscript denotes the degrees of freedom for the numerator and the second value is the degrees of freedom for the denominator. For example, 𝐹 3, 16 represents 3 degrees of freedom for the numerator and 16 degrees of freedom for the denominator. The level 𝛼 critical value of the F distribution is the value for which the area in the tail to the right of the critical value is 𝛼. We will denote the level 𝛼 critical value by 𝑓 𝛼 . We will not need to find critical values for left tails, because the test statistic is the quotient of the larger sample variance by the smaller sample variance. In other words, the value of the test statistic is never less than 1. Table A.5 can be used to find critical values associated with the F distribution.

Example Find 𝑓 0.05 , the 𝛼 = 0.05 critical value, for the 𝐹 3, 16 distribution. Solution: In Table A.5, we find the column corresponding to 3 degrees of freedom for the numerator, and the rows corresponding to 16 degrees of freedom for the denominator. We choose the row for which the tail area is 0.05. The critical value is 𝑓 0.05 = 3.24.

Perform a hypothesis test for two population standard deviations Objective 2 Perform a hypothesis test for two population standard deviations

Critical Region for F distribution The critical region for a one-tailed (either right- or left-tailed) is F ≥ 𝑓 𝛼 . The critical region for a two-tailed test differs from that of other tests. For most other two-tailed tests, the critical region consists of an area of 𝛼 2 in both left and right tails. For the F-test, however, the test statistic never falls into the left tail, so the critical region for a two-tailed test consists of only the right tail. We reject 𝐻 0 when F ≥ 𝑓 𝛼 2 .

Performing a Hypothesis Test for Two Standard Deviations Step 1: State the null and alternate hypotheses. Step 2: Choose a significance level 𝛼 and find the critical value, as follows: One-Tailed: The critical value is 𝑓 𝛼 . Two-Tailed: The critical value is 𝑓 𝛼 2 . Step 3: Compute the test statistic 𝐹= larger of 𝑠 1 2 and 𝑠 2 2 smaller of 𝑠 1 2 and 𝑠 2 2 Step 4: Determine whether to reject 𝐻 0 , as follows: One-Tailed: Reject if F ≥ 𝑓 𝛼 Two-Tailed: Reject if F ≥ 𝑓 𝛼 2 Step 5. State a conclusion.

Example In a series of experiments to determine the absorption rate of certain pesticides into skin, two pesticides were applied to several skin specimens. After a time, the amounts absorbed (in micrograms) were measured. For pesticide 1, the variance of the amounts absorbed in 6 specimens was 2.3, while for pesticide 2, the variance of the amounts absorbed in 10 specimens was 0.6. Assume that for each pesticide, the amounts absorbed are a simple random sample from a normal population. Can we conclude that the standard deviation of the amount absorbed is greater for pesticide 1 than for pesticide 2? Use the 𝛼 = 0.05 significance level. Solution: Because we are interested in determining whether 𝜎 1 > 𝜎 2 , the hypotheses are 𝐻 0 : 𝜎 1 = 𝜎 2 𝐻 1 : 𝜎 1 > 𝜎 2

Solution To find the critical value, we first find the degrees of freedom. The first is the degrees of freedom for the numerator of the test statistic and the second is the degrees of freedom for the denominator of the test statistic. The larger of the variances is 2.3 for a sample of size 6. Therefore, there are 6 – 1 = 5 degrees of freedom for the numerator. The smaller of the variances is 0.6 for a sample of size 10. Therefore, there are 10 – 1 = 9 degrees of freedom for the denominator. Since this is a one-tailed test with 5 and 9 degrees of freedom, the critical value is 𝑓 0.05 = 3.48. The larger sample variance is 2.3 and the smaller one is 0.6. The test statistic is 𝐹= 2.3 0.6 =3.83 Because F > 𝑓 0.05 , we reject 𝐻 0 at the 𝛼 = 0.05 level. We conclude that the standard deviation of the amount absorbed is greater for pesticide 1 than for pesticide 2.

Hypothesis Testing on the TI-84 PLUS The 2-SampFTest command will perform a hypothesis test about two population standard deviations. This command is accessed by pressing STAT and highlighting the TESTS menu. If the summary statistics are given the Stats option should be selected for the input option. If the raw sample data are given, the Data option should be selected. Note that the input is the sample standard deviation of each sample. You may be given the variance in the problem.

Example (TI-84 PLUS) In a series of experiments to determine the absorption rate of certain pesticides into skin, two pesticides were applied to several skin specimens. After a time, the amounts absorbed (in micrograms) were measured. For pesticide 1, the variance of the amounts absorbed in 6 specimens was 2.3, while for pesticide 2, the variance of the amounts absorbed in 10 specimens was 0.6. Assume that for each pesticide, the amounts absorbed are a simple random sample from a normal population. Can we conclude that the standard deviation of the amount absorbed is greater for pesticide 1 than for pesticide 2? Use the 𝛼 = 0.05 significance level. Solution: We press STAT and highlight the TESTS menu and select 2-SampFTest.

Example (TI-84 PLUS) We are given 𝑠 1 2 =2.3, 𝑛1 = 6, 𝑠 2 2 =0.6, and n1 =10. Since the inputs for the 2-SampFTest procedure are the sample standard deviations, we enter 𝑠 1 = 2.3 and 𝑠 2 = 0.6 . Since we have a right-tailed test, select the > 𝝈 𝟐 option. Select Calculate. The P-value is 0.3876. Since this is less than the significance level 𝛼 = 0.05, we reject 𝐻 0 . We conclude that the standard deviation of the amount absorbed is greater for pesticide 1 than for pesticide 2.

F test is Sensitive to the Normality Assumption The F-test, like the 𝑡-test, requires that the samples come from normal populations. Unlike the 𝑡-test, the F-test for comparing variances is very sensitive to this assumption. If the shapes of the populations differ much from the normal curve, the F-test may give misleading results. For this reason, the F-test for comparing variances must be used with caution.

You Should Know… How to find critical values of the F distribution How to perform a hypothesis test for two population standard deviations