Aim How do we test the difference between

Aim: How do we test the difference between two variances? HW#13: complete two questions on last slides

Comparing Data • Not only do we test the difference between two means (yesterdays aim) but statisticians are interested in comparing t two variances and standard deviations

What test do we use? • For comparison of two variances or standard deviations, an F test is used

Characteristics of the F Distribution 1. The values of F cannot be negative, because variances are always positive or zero 2. The distribution is positively skewed 3. The mean value of F is approximately equal to 1 4. The F distribution is a family of curves based on the degrees of freedom of the variance of the numerator and the degrees of freedom of the variance of the denominator

Shapes of F Distributions

Formula for the F Test • The larger of the two variances is placed in the numerator regardless of the subscripts

Degrees of Freedom • d. f. N degree of freedom for numerator • d. f. D degree of freedom for denominator • NEED TO USE H TABLE

Example • Find the critical value for a right-tailed F test when α=0. 05, the degrees of freedom for the numerator are 15, and the degrees of freedom for the denominator are 21. • Since the test is right tailed with α=0. 05, use the 0. 05 table. The d. f. N is listed across the top and the d. f. D is listed in the left column – Solution: 2. 18

Using F table • When the degree of freedom values cannot be found in the table, closest values on the smaller side should be used – Example: If d. f. N = 14, this value is between given table values of 12 and 15; therefore 12 should be used, to be the safe side

Testing the equality of two variances Left-Tailed Right-Tailed Two-Tailed

Steps 1. 2. 3. 4. 5. State the hypothesis and identify claim Find the critical value Compare the test values Make the decision Summarize the results

Example • A medical researcher wishes to see whether the variance of the heart rates (in beats per minute) of smokers is different from the variance of heart rates of people, who do not smoke. Two samples are selected, and the data are as shown. Using α=0. 05, is there enough evidence to support the claim? Smoker Nonsmoker n = 26 n = 18 s 2 = 36 s 2 = 10

Solution Critical value from table = 2. 56 F = 36/10 = 3. 6 > 2. 56 Summarize the results: There is enough evidence to support the claim that the variance of the heart rates of smokers and nonsmokers is different

Class Work #2 1. Using the table H, find the P-value interval for each FTest value 1. 2. F = 2. 97, d. f. N=9, d. f. D=14, right tailed F=3. 32, d. f. N=6, d. f. D=12, two tailed A researcher claims that the standard deviation of the ages of cats is smaller than the standard deviation of the ages of dogs who are owned by families in a large city. A randomly selected sample of 29 cats has a standard deviation of 2. 7 years and a random sample of 16 dogs has a standard deviation of 3. 5 years. Is the researcher correct? Use α=0. 05. If there is a difference, suggest a reason for the difference.

Homework Question 1

Homework Question 2