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11- 1 Chapter Eleven McGraw-Hill/Irwin © 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Chapter Eleven 11- 2 Two-Sample Tests of Hypothesis GOALS When you have completed this chapter, you will be able to: ONE Conduct a test of hypothesis about the difference between two independent population means. TWO Conduct a test of hypothesis regarding the difference in two population proportions. THREE Conduct a test of hypothesis about the mean difference between paired or dependent observations. Chapter Eleven continued Two Sample Tests of Hypothesis GOALS When you have completed this chapter, you will be able to: FOUR Understand the difference between dependent and independent samples. 11- 3 11- 4 Comparing two populations Does the distribution of the differences in sample means have a mean of 0? If both samples contain at least 30 observations we use the z distribution as the test statistic. The samples are from independent populations. No assumptions about the shape of the populations are required. The formula for X1 X 2 z computing the s12 s 22 value of z is: n1 n2 Comparing two populations 11- 5 Two cities, Bradford and Kane are separated only by the Conewango River. There is competition between the two cities. The local paper recently reported that with a standard deviation the mean household income of $7,000 for a sample of in Bradford is $38,000 with 35 households. At the .01 a standard deviation of significance level can we $6,000 for a sample of 40 conclude the mean income households. The same in Bradford is more? article reported the mean income in Kane is $35,000 EXAMPLE 1 11- 6 Step 4 State the decision rule. The null hypothesis is rejected if z is greater than 2.33 or p < .01. Step 1 State the null and alternate hypotheses. H0: µB < µK H1: µB > µK Step 3 Find the appropriate test statistic. Because both samples are more than 30, we can use z as the test statistic. Step 2 State the level of significance. The .01 significance level is stated in the problem. Example 1 continued 11- 7 Step 5: Compute the value of z and make a decision. z The p(z > 1.98) is .0239 for a one-tailed test of significance. $38,000 $35,000 ($6,000 ) 2 ($7,000 ) 2 40 35 1.98 Because the computed Z of 1.98 < critical Z of 2.33, the p-value of .0239 > a of .01, the decision is to not reject the null hypothesis. We cannot conclude that the mean household income in Bradford is larger. Example 1 continued 11- 8 Small Sample Tests of Means The t distribution is used as the test statistic if one or more of the samples have less than 30 observations. The required assumptions 1. Both populations must follow the normal distribution. 2. The populations must have equal standard deviations. 3. The samples are from independent populations. Small Sample Tests of Means 11- 9 Finding the value of the test statistic requires two steps. Step One: Pool the sample standard deviations. 2 2 ( n 1 ) s ( n 1 ) s 1 2 2 s 2p 1 n1 n2 2 Step Two: Determine the value of t from the following formula. t X1 X 2 2 s p 1 1 n1 n2 Small sample test of means continued 11- 10 A recent EPA study compared the highway fuel economy of domestic and imported passenger cars. A sample of 15 domestic cars revealed a mean of 33.7 mpg with a standard deviation of 2.4 mpg. A sample of 12 imported cars revealed a mean of 35.7 mpg with a standard deviation of 3.9. At the .05 significance level can the EPA conclude that the mpg is higher on the imported cars? Example 3 11- 11 Step 3 Find the appropriate test statistic. Both samples are less than 30, so we use the t distribution. Step 1 State the null and alternate hypotheses. H0: µD > µI H1: µD < µI Step 2 State the level of significance. The .05 significance level is stated in the problem. Example 3 continued 11- 12 Step 4 The decision rule is to reject H0 if t<-1.708 or if p-value < .05. There are n-1 or 25 degrees of freedom. Step 5 We compute the pooled variance. 2 2 ( n 1 )( s ) ( n 1 )( s 2 1 1 2 2) sp n1 n 2 2 (15 1)( 2.4) 2 (12 1)(3.9) 2 9.918 15 12 2 Example 3 continued 11- 13 We compute the value of t as follows. t X1 X 2 1 1 n n 2 1 33 .7 35 .7 s 2p 1 1 8.312 15 12 1.640 Example 3 continued 11- 14 P(t < -1.64) = .0567 for a onetailed t-test. Since a computed z of –1.64 > critical z of –1.71, the pvalue of .0567 > a of .05, H0 is not rejected. There is insufficient sample evidence to claim a higher mpg on the imported cars. Example 3 continued Example 2 11- 15 11- 16 Test the hypothesis that there is no significant difference in the mean mounting times between the two procedures. 11- 17 Level of significance: a=0.10 Test Statistics: Independent Samples t test Decision Rule: Reject Ho if t computed > 1.833 Computations: 11- 18 Decision: Fail to reject the null hypothesis Conclusion: There is no significant difference in the mean mounting times between the two procedures. Independent samples are samples that are not related in any way. 11- 19 Dependent samples are samples that are paired or related in some fashion. If you wished to buy a car you would look at the same car at two (or more) different dealerships and compare the prices. If you wished to measure the effectiveness of a new diet you would weigh the dieters at the start and at the finish of the program. Hypothesis Testing Involving Paired Observations 11- 20 Use the following test when the samples are dependent: d t sd / n where d is the mean of the differences sd is the standard deviation of the differences n is the number of pairs (differences) Hypothesis Testing Involving Paired Observations An independent testing agency is comparing the daily rental cost for renting a compact car from Hertz and Avis. A random sample of eight cities revealed the following information. At the .05 significance level can the testing agency conclude that there is a difference in the rental charged? 11- 21 City Atlanta Chicago Cleveland Denver Honolulu Kansas City Miami Seattle Hertz ($) 42 56 45 48 37 45 41 46 Avis ($) 40 52 43 48 32 48 39 50 EXAMPLE 4 11- 22 Step 1 Ho: md = 0 H1: md = 0 Step 2 The stated significance level is .05. Step 3 The appropriate test statistic is the paired t-test. Step 4 H0 is rejected if t < -2.365 or t > 2.365; or if p-value < .05. We use the t distribution with n-1 or 7 degrees of freedom. Step 5 Perform the calculations and make a decision. Example 4 continued 11- 23 City Hertz Avis d d2 Atlanta 42 40 2 4 Chicago 56 52 4 16 Cleveland 45 43 2 4 Denver 48 48 0 0 Honolulu 37 32 5 25 Kansas City 45 48 -3 9 Miami 41 39 2 4 Seattle 46 50 -4 16 Example 4 continued 11- 24 d 8.0 d 1.00 n 8 2 d d 2 n sd n 1 d t sd n 82 78 8 3.1623 8 1 1.00 3.1623 0.894 8 Example 4 continued 11- 25 P(t>.894) = .20 for a one-tailed t-test at 7 degrees of freedom. Because 0.894 is less than the critical value, the p-value of .20 > a of .05, do not reject the null hypothesis. There is no difference in the mean amount charged by Hertz and Avis. Example 4 continued 11- 26 Problem 1 11- 27 Problem 2 11- 28 Problem 3 Problem 4 11- 29