Hypothesis Testing – Two Population Variances Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. HAWKES LEARNING SYSTEMS math courseware specialists Section 11.5 Hypothesis Testing – Two Population Variances

The two populations must be normally distributed. HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing (Two or More Populations) 11.5 Hypothesis Testing – Two Population Variances Criteria for hypothesis testing with variances: The two populations must be independent, not matched or paired in any way. The two populations must be normally distributed.

The F-distribution is skewed to the right. HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing (Two or More Populations) 11.5 Hypothesis Testing – Two Population Variances Critical Value, F: The F-distribution is skewed to the right. The values of F are always greater than 0. The shape of the F-distribution is completely determined by its two parameters, the degrees of freedom of the numerator and the degrees of freedom of the denominator of the ratio. In this section, in order to be consistent with the F-distribution tables, we will round calculated values to four decimal places.

HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing (Two or More Populations) 11.5 Hypothesis Testing – Two Population Variances Test Statistic for Population Variances: F = To determine if the test statistic calculated from the sample is statistically significant we will need to look at the critical value. The critical values for population variances are found from the f-distribution table. The f-distribution table assumes that the area is to the right of f-critical.

Reject if F ≤ F1 –  HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing (Two or More Populations) 11.5 Hypothesis Testing – Two Population Variances Rejection Regions for Left-Tailed Tests, Ha contains <: The f-distribution table assumes that the area is to the right of f-critical. Obtain the critical value F1 –  Reject if F ≤ F1 – 

Reject if F ≥ F HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing (Two or More Populations) 11.5 Hypothesis Testing – Two Population Variances Rejection Regions for Right-Tailed Tests, Ha contains >: The f-distribution table assumes that the area is to the right of f-critical. Obtain the critical value F Reject if F ≥ F

Reject if F ≤ or F ≥ HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing (Two or More Populations) 11.5 Hypothesis Testing – Two Population Variances Rejection Regions for Two-Tailed Tests, Ha contains ≠: Reject if F ≤ or F ≥

State the null and alternative hypotheses. HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing (Two or More Populations) 11.5 Hypothesis Testing – Two Population Variances Steps for Hypothesis Testing: State the null and alternative hypotheses. Set up the hypothesis test by choosing the test statistic and determining the values of the test statistic that would lead to rejecting the null hypothesis. Gather data and calculate the necessary sample statistics. Draw a conclusion.

HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing (Two or More Populations) 11.5 Hypothesis Testing – Two Population Variances Null and Alternative Hypotheses:

First state the hypotheses: H0: Ha: HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing (Two or More Populations) 11.5 Hypothesis Testing – Two Population Variances Draw a conclusion: A quality control inspector believes that the machines on Assembly Line A are not adjusted properly and that the variance in the size of the candy produced is greater than the variance in the size of candy produced by Assembly Line B. A sample of 20 pieces of candy is taken from each assembly line and measured. The size of the candy from Assembly Line A has a sample variance of 1.45, while the size of the candy from Assembly Line B has a sample variance of 0.47. Using a 0.01 level of significance, perform a hypothesis test to test the quality control inspector’s claim. Solution: First state the hypotheses: H0: Ha: Next, set up the hypothesis test and state the level of significance:  = 0.01, d.f. = 19 F = Reject if F ≥ 3.0274. 3.0274

Gather the data and calculate the necessary sample statistics: HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing (Two or More Populations) 11.5 Hypothesis Testing – Two Population Variances Solution (continued): Gather the data and calculate the necessary sample statistics: = 1.45, = 0.47 Finally, draw a conclusion: Since F-statistic (3.0851) is greater than F (3.0274), we will reject the null hypothesis. The evidence does sufficiently support the inspector’s claim. F = 3.0851

First state the hypotheses: H0: Ha: HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing (Two or More Populations) 11.5 Hypothesis Testing – Two Population Variances Draw a conclusion: A professor claims that the variances in the test scores on two different versions of an exam are not equal. To test the claim, a sample of 20 scores of version A is chosen and a sample variance of 3.815 is found, while a sample of 20 scores of version B has a sample variance of 3.391. Conduct a hypothesis test to test the professor’s claim that the variances of the test scores are not equal. Use a 0.05 level of significance. Solution: First state the hypotheses: H0: Ha: Next, set up the hypothesis test and state the level of significance:  = 0.05, d.f. = 19 = F0.025 = 2.5265 and = F0.975 = 0.3958 Reject if F ≤ 0.3958 or F ≥ 2.5265.

Gather the data and calculate the necessary sample statistics: HAWKES LEARNING SYSTEMS math courseware specialists Hypothesis Testing (Two or More Populations) 11.5 Hypothesis Testing – Two Population Variances Solution (continued): Gather the data and calculate the necessary sample statistics: = 3.815, = 3.391 Finally, draw a conclusion: Since F-statistic (1.1250) is not in the rejection region(F is between 0.3958 and 2.5265), we will fail to reject the null hypothesis. The evidence does not sufficiently support the claim that the two tests have different variances. F = 1.1250 Ho: Ha:

The table assumes that the area is to the right of f-critical. Thus, alpha of 0.1 for right tail test corresponds to alpha of 0.9 for left tail tests.  = 0.9 DF1 = 19 and DF2 = 40

F Statistic = Reject if F < F0.9 or p-value > (1- ) Reject if F < 0.5781 or p-value > 0.9 Computed- (F-Statistic) is 0.5028 and it is less than 0.5781 p-value is 0.94545 and it is greater than 0.9 Reject the Null Hypothesis. There is enough evidence that σ12 is lower than σ22

The table assumes that the area is to the right of f-critical. DF1 = 17 and DF2 = 15 The table assumes that the area is to the right of f-critical. But, this is a two-tail test and alpha belongs to both tails. Thus, alpha of 0.01 for two-tail tests corresponds to: - alpha of 0.005 for the right tail  (/2)  3.9827 - alpha of 0.995 for the left tail  (1- /2)  0.2636

F Statistic = Reject if F > F0.005 or F < F0.995 or p-value < (/2) OR p-value > (1- /2) Reject if F > 3.9827 or F < 0.2636 or p-value < 0.005 OR p-value > 0.995 Computed- (F-Statistic) is 0.4530 and it is NOT > 3.9827 OR < 0.2636 p-value is 0.94058 and it is not less than 0.005 or greater than 0.995 Don’t reject the Null Hypothesis. There is NOT enough evidence that they are different.

F Statistic = = 0.02 = 3.333 0.006

Reject the Null Hypothesis. There is enough evidence that σ12 is greater than σ22