Slide 1

Section 8.2 Testing the Difference Between Means (Small Independent Samples) (Independent Samples, 1 and 2 Unknown)

Slide 2

Section 8.2 Objectives Perform a t-test for the difference between two population means 1 and 2 using independent samples with population standard deviations unknown.

Slide 3

Two Sample t - Test for the Difference Between Means These conditions are necessary to use a t-test for the difference between means for independent samples. 1.The samples must be randomly selected. 2.The samples must be independent. 3.Each population must have a normal distribution -or- each sample size is at least 30. 4.The population standard deviations ( 1, 2 ) are UNKNOWN

Slide 4

Two Sample t - Test for the Difference Between Means The test statistic is. The standardized test statistic is The standard error and the degrees of freedom of the sampling distribution depend on whether the population variances and are equal.

Slide 5

The standard error for the sampling distribution of is Two Sample t - Test for the Difference Between Means Variances are equal Information from the two samples is combined to calculate a pooled estimate of the standard deviation. d.f.= n 1 + n 2 2

Slide 6

Variances are not equal If the population variances are not equal, then the standard error is d.f = smaller of n 1 1 and n 2 1 Two Sample t - Test for the Difference Between Means

Slide 7

Normal or t-Distribution? Both populations normally distributed or are both sample sizes at least 30? You cannot use the z- test or the t-test. (!) Yes Are both population standard deviations ( 1 and 2 ) known? No Are the population variances equal? Use the z-test. Use the t-test with d.f = smaller of n 1 1 or n 2 1. Use the t-test with Yes No Yes d.f = n 1 + n 2 2.

Slide 8

Two-Sample t-Test for the Difference Between Means (Independent Samples, 1, 2 Unknown) 1.State the claim mathematically and verbally. Identify the null and alternative hypotheses. 2.Specify the level of significance. 3.Determine the degrees of freedom. 4.Determine the critical value(s). State H 0 and H a. Identify . Use Table 5 in Appendix B. d.f. = n 1 + n 2 2 (equal ) or d.f. = smaller of n 1 1 or n 2 1. (unequal ) In WordsIn Symbols

Slide 9

Two-Sample t-Test for the Difference Between Means (Small Independent Samples) 5.Determine the rejection region(s). 6.Find the standardized test statistic and sketch the sampling distribution. 7.Make a decision to reject or fail to reject the null hypothesis. 8.Interpret the decision in the context of the original claim. If t is in the rejection region, reject H 0. Otherwise, fail to reject H 0. In WordsIn Symbols

Slide 10

Example: Two - Sample t - Test for the Difference Between Means The results of a state mathematics test for random samples of students taught by two different teachers at the same school are shown below. Can you conclude that there is a difference in the mean mathematics test scores for the students of the two teachers? Use = 0.10. Assume the populations are normally distributed, and the population variances are not equal and they are unknown. Teacher 1Teacher 2 s 1 = 39.7s 2 = 24.5 n 1 = 8n 2 = 18

Slide 11

Solution: Two - Sample t - Test for the Difference Between Means H 0 : H a : d.f. = Rejection Region: Test Statistic: 0.10 8 1 = 7 1 = 2 1 2 (claim) Decision: At the 10% level of significance, there is not enough evidence to support the claim that the mean mathematics test scores for the students of the two teachers are different. Fail to Reject H 0. t 0.922

Slide 12

Example: Two - Sample t - Test for the Difference Between Means A manufacturer claims that the mean calling range (in feet) of its 2.4-GHz cordless telephone is greater than that of its leading competitor. You perform a study using 14 randomly selected phones from the manufacturer and 16 randomly selected similar phones from its competitor. The results are shown below. At = 0.05, can you support the manufacturers claim? Assume the populations are normally distributed and the population variances are equal (and unknown). ManufacturerCompetitor s 1 = 45 fts 2 = 30 ft n 1 = 14 n 2 = 16

Slide 13

Solution: Two - Sample t - Test for the Difference Between Means H 0 : H a : = d.f. = Rejection Region: 0.05 14 + 16 2 = 28 1 2 1 > 2 (claim) t 01.701 0.05

Slide 14

Solution: Two - Sample t - Test for the Difference Between Means Test Statistic:

Slide 15

Solution: Two - Sample t - Test for the Difference Between Means H 0 : H a : = d.f. = Rejection Region: Test Statistic: 0.05 14 + 16 2 = 28 1 2 1 > 2 (claim) 1.811 Decision: At the 5% level of significance, there is enough evidence to support the manufacturers claim that its phone has a greater mean calling range than its competitor. Reject H 0. t 01.701 0.05

Slide 16

The maximal oxygen consumption is a way to measure the fitness of an individual. It is the amount of oxygen in milliliters a person uses per kilogram of body weight per minute. A medical research center claims that athletes have a greater mean maximal oxygen consumption than non-athletes. The results for samples from the 2 groups are below. At = 0.05, can you support the research centers claim? (Assume population variances are equal, and populations are normally distributed, s unknown.) AthletesNon-athletes s 1 = 4.9 ml/kg/mins 2 = 3.1 ml/kg/min n 1 = 23n 2 = 21

Slide 17

Test of Inference I. H o : 1 2 II. = 0.05 H a : 1 > 2 (claim) III. IV. V. VI.

Slide 18

Test of Inference

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Section 8.2 Summary Performed a t-test for the difference between two means 1 and 2 from independent samples when the population standard deviations are not known.