CHAPTER 9 Testing the Difference Between Two Means Two Variances and Two Proportions Student Notes Stat 200 Elementary Statistics Introduction Test Test Test Test Test the the the the the difference difference difference difference difference between between between between between two two two two two large sample means using the z test variances or standard deviations means for small independent samples means for small dependent samples proportions Section 9 1 Testing the Difference Between Two Means Using the z Test Hypotheses The Difference Between Two Means Assumptions 1 The samples must be independent of each other that is there can be no relationship between the subjects in each sample 2 The populations from which the samples were obtained must be normally distributed and the standard deviations of the variable must be known or the sample sizes must be greater than or equal to 30 Formula z X1 X 2 1 2 12 22 n1 n 2 Example 1 A survey found that the average hotel room rate in New Orleans is 88 42 and the average room rate in Phoenix is 80 61 Assume that the data were obtained from two samples of 50 hotels each and that the standard deviations for the population were 5 62 and 4 83 respectively At a 0 05 can it be concluded that there is a significant difference in the rates Example 2 A researcher hypothesizes that the average number of sports that colleges offer for males is greater than the average number of sports that colleges offer for females A sample of the number of sports offered by colleges is shown At a 0 10 is there enough evidence to support the claim Use the P value method 6 11 6 9 15 6 8 9 7 7 11 5 11 5 5 Males 8 15 6 14 6 9 6 9 5 5 9 9 11 6 9 5 10 7 10 7 8 12 18 7 5 5 11 5 10 8 18 6 8 8 11 6 6 16 9 11 8 11 13 5 5 7 10 7 8 18 13 7 4 6 8 Females 8 7 5 6 10 7 5 7 5 10 7 8 7 14 12 13 6 5 5 5 Confidence Interval for the Difference Between Two Means X 1 X 2 z 2 12 22 2 2 1 2 X 1 X 2 z 2 1 2 n1 n2 n1 n2 Example 3 A survey found that the average hotel room rate in New Orleans is 88 42 and the average room rate in Phoenix is 80 61 Assume that the data were obtained from two samples of 50 hotels each and that the standard deviations were 5 62 and 4 83 respectively Find the 95 confidence interval for the difference between the means Try this Two groups of students are given a problem solving test and the results are compared Find the 90 confidence interval of the true difference in means Msthematics majors Computer science majors mean1 83 6 mean2 79 2 s1 4 3 s2 3 8 n1 36 n2 36 14 5 6 7 8 6 5 5 6 5 Section 9 2 Testing the Difference Between Two Means of Independent Samples Using the t Test t Test t test is used to test the difference between means when the two samples are independent when the sample sizes are small and when the samples are taken from two normally or approximately normally distributed populations There are two different options for the use of t tests One option is used when the variances of the populations are not equal and the other option is used when the variances are equal A Difference Between Two Means Small Samples When variances are assumed to be unequal t X1 X 2 1 2 s12 s22 n1 n 2 where the degrees of freedom are equal to the smaller of n1 1 or n2 1 When variances are assumed to be equal t X 1 X 2 1 2 n1 1 s12 n2 1 s22 n1 n2 2 1 1 n1 n2 where the degrees of freedom are equal to n1 n2 2 Example 1 The average size of a farm in Indiana County Pa is 191 acres The average size f a farm in Greene County Pa is 199 acres Assume the data were obtained from two samples with standard deviations of 38 and 12 acres respectively and sample sizes 8 and 10 respectively Can it be concluded at a 0 05 that the average size of the farms in the two counties is different Assume the populations are normally distributed Example 2 A researcher wishes to determine whether the salaries of professional nurses employed by private hospitals are higher than those of nurses employed by government owned hospitals She selects a sample of nurses from each type of hospital and calculates the means and standard deviations of their salaries At 0 01 can she conclude that the private hospitals pay more than the government hospitals Assume that the populations are approximately normally distributed Use the p value method Confidence Intervals Unequal Variance d f the smaller value of n1 1 or n2 1 Confidence Intervals Equal Variance X 1 X 2 t 2 n1 1 s12 n2 1 s22 n1 n2 2 1 2 X 1 X 2 t 2 1 1 n1 n2 n1 1 s12 n2 1 s22 n1 n2 2 1 1 n1 n2 d f n1 n2 2 Example 3 The average size of a farm in Indiana County Pa is 191 acres The average size of a farm in Greene County Pa is 199 acres Assume the data were obtained from two samples with standard deviations of 38 and 12 acres respectively and sample sizes 8 and 10 respectively Assume the populations are normally distributed Find the 95 confidence interval 9 2 Testing the Difference Between Two Means Dependent Samples General Procedure Finding the Test Value Step 1 Step 2 Step 3 Step 4 Step 5 Find the differences of the values of the pairs of data D Find the mean of the differences Find the standard deviation of the differences Find the estimated standard error of the differences Find the test value t t Test Formulas D D t sD n D D n sD D 2 2 D n 1 n Example 1 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 a 0 05 Assume that the variable is approximately normally distributed Example 2 A dietitian 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 Can it be concluded that the cholesterol level has been changed at a 0 10 Assume that the variable …