Chapter 9

 Many instances when researchers wish to compare two sample means  Examples: ◦ Average lifetimes of two different brands of bus tires ◦ Two different brands of fertilizer to see which is better for growing plants ◦ Two brands of cough syrup to see which is more effective

 In comparison of two means, same basic steps for hypothesis testing are used ◦ z test and t test are both used  When using the t test, researcher must decide if two samples are independent or dependent  z test can be used to test two proportions  F test is used to test two variances

 Assumptions for the Test to Determine the Difference Between Two Means 1.Samples must be independent of each other. There is no relationship between subjects in each sample. 2.Standard deviation of both populations must be known, and if sample sizes are less than 30, populations must be normally or approx. normally distributed.

 Theory behind testing difference between two means is based on selecting pairs of samples and comparing means of the pairs ◦ Population means need not be known ◦ All possible pairs are taken from population ◦ Means of each pair are computed, subtracted, then plotted ◦ Curve of the plotted differences will be shaped similar to the normal curve

 A survey found that the average hotel room rate in New Orleans is $88.42 and the average room rate in Phoenix is $ Assume the data were obtained from two samples of 50 hotels each and that the standard deviations of the populations were $5.62 and $4.83, respectively. At α = 0.05, can it be concluded that there is a significant difference in the rates?

 A researcher hypothesizes that the average number of sports that colleges offer for males is greater than the average number of sports colleges offer for females. A sample of the number of sports offered by colleges is shown (page 477). At α = 0.10, is there enough evidence to support the claim? Use the P-value method, and assume both population standard deviations are equal to 3.3.

 Find the 95% confidence interval for the difference between the means for the data in example 9 – 1.

 When population standard deviations are unknown, a t test is used to test difference between two means  Two samples must be independent and taken from two normally or approx. normally distributed populations  Independent samples ◦ Samples are not related

 The average size of a farm in Indiana County, Pennsylvania, is 191 acres. The average size of a farm in Greene County, Pennsylvania, is 199 acres. Assume the data were obtained from two samples with standard deviations of 38 and 12 acres, respectively, and sample sizes of 8 and 10, respectively. Can it be concluded at α = 0.05 that the average size of the farms in the two counties is different? Assume the populations are normally distributed.

 Find the 95% confidence interval for the data in example 9 – 4

 Different version of t test is used when samples are dependent  Dependent samples ◦ Subjects of samples are paired or matched in some way  In a pre-test and post-test situation, only a gain or loss in values is compared

 A physical education director claims by taking a special vitamin, a weight lifter can increase his strength. Eight athletes are selected and given a test of strength, using the standard bench press. After 2 weeks of regular training, supplemented with the vitamin, they are tested again. Test the effectiveness of the vitamin regimen at α = Each value in these data represents the maximum number of pounds the athlete can bench-press. Assume that the variable is approximately normally distributed.

 A dietician wishes to see if a person’s cholesterol level will change if the diet is supplemented by a certain mineral. Six subjects were pretested, and then they took the mineral supplement for a 6-week period. The results are shown in the table (page 495). Can it be concluded that the cholesterol level has been changed at α = 0.10? Assume the variable is approximately normally distributed.

1. State hypotheses and identify claim 2. Find critical value(s) 3. Compute test value ◦ (by hand or with calculator using lists) 4. Make decision 5. Summarize results

 z test can also be used to test equality of two proportions  Example: ◦ Is proportion of men who exercise regularly less than proportion of women who exercise regularly? ◦ Is there a difference in proportion of college grads who pay cash for purchases and proportion of non- college grads who pay cash?

 Two requirements for use of z test for proportions: 1.Samples must be independent 2. n 1 p 1 and n 1 q 1 must be 5 or more n 2 p 2 and n 2 q 2 must be 5 or more

 In a nursing home study, researchers found that 12 of 34 small nursing home had a resident vaccination rate of less than 80%, while 17 out of 24 large nursing homes had a vaccination rate of less than 80%. At α = 0.05, test the claim that there is no difference in the proportions of the small and large nursing homes with a resident vaccination rate of less than 80%.

 In a sample of 200 workers, 45% said that they missed work because of personal illness. Ten years ago in a sample of 200 workers, 35% said that they missed work because of personal illness. At α = 0.01, is there a difference in the proportion?