CHAPTER 9 Testing the Difference Between Two Means, Two Variances, and Two ProportionsStat 200 – Elementary StatisticsIntroductionHypothesesExample 3Example 2A 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.General Procedure—Finding the Test ValueStep 1 Find the differences of the values of the pairs of data, D.Step 2 Find the mean of the differences .Step 3 Find the standard deviation of the differences, .Step 4 Find the estimated standard error of the differences, .Step 5 Find the test value, t.The Difference Between Two ProportionsThe confidence interval for the difference between two proportions can be calculated using the following formula:Stat 200 – Elementary Statistics IntroductionTest the difference between two large sample means using the z test.Test the difference between two variances or standard deviations.Test the difference between two means for small independent samples.Test the difference between two means for small dependent samples.Test the difference between two proportions.Section 9.1 Testing the Difference Between Two Means: Using the z TestHypothesesThe Difference Between Two Means Assumptions1. 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.FormulaExample 1A 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?Student NotesCHAPTER 9 Testing the Difference Between Two Means, Two Variances, and Two ProportionszX Xn n  ( ) ( )1 2 1 2121222 Example 2A 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 supportthe claim? (Use the P-value method)Confidence Interval for the Difference Between Two MeansExample 3A 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 Males6 11 11 8 15 6 14 8 12 186 9 5 6 9 6 9 18 7 615 6 11 5 5 9 9 5 5 88 9 5 11 6 9 5 11 5 87 7 5 10 7 10 7 10 8 11 Females6 8 11 13 8 7 5 13 14 66 5 5 7 6 10 7 6 5 516 10 7 8 5 7 5 5 6 59 18 13 7 10 7 8 5 7 611 4 6 8 7 14 12 5 8 5   2221212/21212221212/21nnzXXnnzXXSection 9.2 Testing the Difference Between Two Means of Independent SamplesUsing the t Testt TestA 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 isused when the variances of the populations are not equal, and the other option is used when the variances are equal.Difference Between Two Means–Small SamplesWhen variances are assumed to be unequal: where the degrees of freedom are equal to the smaller of n1 – 1 or n2 – 1.When variances are assumed to be equal: where the degrees of freedom are equal to n1 + n2 – 2.tX Xsnsn  ( ) ( )1 2 1 2121222   21212222112121112)1()1(nnnnsnsnXXtExample 1The 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 2A 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 d.f. = n1 + n2 – 2 Example 3The 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,