Chapter 10 Comparisons Involving Means Part A • Estimation of the Difference between the Means of Two Populations: Independent Samples • Hypothesis Tests about the Difference between the Means of Two Populations: Independent Samples Estimation of the Difference Between the Means of Two Populations: Independent Samples • Point Estimator of the Difference between the Means of Two Populations • Sampling Distribution of x1 x2 • Interval Estimate of Large-Sample Case • Interval Estimate of Small-Sample Case Point Estimator of the Difference Between the Means of Two Populations Let 1 equal the mean of population 1 and 2 equal the mean of population 2. The difference between the two population means is 1 - 2. To estimate 1 - 2, we will select a simple random sample of size n1 from population 1 and a simple random sample of size n2 from population 2. Let x1 equal the mean of sample 1 and x2 equal the mean of sample 2. The point estimator of the difference between the means of the populations 1 and 2 is x1 x2 . Sampling Distribution of x1 x2 Expected Value E ( x1 x2 ) 1 2 Standard Deviation x1 x2 12 n1 22 n2 where: 1 = standard deviation of population 1 2 = standard deviation of population 2 n1 = sample size from population 1 n2 = sample size from population 2 Interval Estimate of 1 - 2: Large-Sample Case (n1 > 30 and n2 > 30) • Interval Estimate with 1 and 2 Known x1 x2 z / 2 x1 x2 where: 1 - is the confidence coefficient Interval Estimate of 1 - 2: Large-Sample Case (n1 > 30 and n2 > 30) Interval Estimate with 1 and 2 Unknown x1 x2 z / 2 sx1 x2 where: sx1 x2 s12 s22 n1 n2 Interval Estimate of 1 - 2: Large-Sample Case (n1 > 30 and n2 > 30) • Example: Par, Inc. Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.” In a test of driving distance using a mechanical driving device, a sample of Par golf balls was compared with a sample of golf balls made by Rap, Ltd., a competitor. The sample statistics appear on the next slide. Interval Estimate of 1 - 2: Large-Sample Case (n1 > 30 and n2 > 30) • Example: Par, Inc. Sample Size Sample Mean Sample Std. Dev. Sample #1 Par, Inc. 120 balls 235 yards 15 yards Sample #2 Rap, Ltd. 80 balls 218 yards 20 yards Point Estimator of the Difference Between the Means of Two Populations Population 1 Par, Inc. Golf Balls Population 2 Rap, Ltd. Golf Balls 1 = mean driving 2 = mean driving distance of Par golf balls distance of Rap golf balls m1 – 2 = difference between the mean distances Simple random sample of n1 Par golf balls Simple random sample of n2 Rap golf balls x1 = sample mean distance for sample of Par golf ball x2 = sample mean distance for sample of Rap golf ball x1 - x2 = Point Estimate of m1 – 2 Point Estimate of the Difference Between Two Population Means Point estimate of 1 2 = x1 x2 = 235 218 = 17 yards where: 1 = mean distance for the population of Par, Inc. golf balls 2 = mean distance for the population of Rap, Ltd. golf balls 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case, 1 and 2 Unknown Substituting the sample standard deviations for the population standard deviation: x1 x2 z / 2 12 22 (15) 2 ( 20) 2 17 1. 96 n1 n2 120 80 17 + 5.14 or 11.86 yards to 22.14 yards We are 95% confident that the difference between the mean driving distances of Par, Inc. balls and Rap, Ltd. balls is 11.86 to 22.14 yards. Using Excel to Develop an Interval Estimate of 1 – 2: Large-Sample Case 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Formula Worksheet A Par 195 230 254 205 260 222 241 217 228 255 209 251 229 220 B C Rap 226 Sample Size 198 Mean 203 Stand. Dev. 237 235 Confid. Coeff. 204 Lev. of Signif. 199 z Value 202 240 Std. Error 221 Marg. of Error 206 201 Pt. Est. of Diff. 233 Lower Limit 194 Upper Limit D Par, Inc. 120 =AVERAGE(A2:A121) =STDEV(A2:A121) 0.95 =1-D6 =NORMSINV(1-D7/2) =SQRT(D4^2*/D2+E4^2/E2) =D8*D10 =D3-E3 =D13-D11 =D13+D11 Note: Rows 16-121 are not shown. E Rap, Ltd. 80 =AVERAGE(A2:A81) =STDEV(A2:A81) Using Excel to Develop an Interval Estimate of 1 – 2: Large-Sample Case 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Value Worksheet A Par 195 230 254 205 260 222 241 217 228 255 209 251 229 220 B C Rap 226 Sample Size 198 Mean 203 Stand. Dev. 237 235 Confid. Coeff. 204 Lev. of Signif. 199 z Value 202 240 Std. Error 221 Marg. of Error 206 201 Pt. Est. of Diff. 233 Lower Limit 194 Upper Limit D Par, Inc. 120 235 15 0.95 0.05 1.960 2.622 5.139 17 11.86 22.14 Note: Rows 16-121 are not shown. E Rap, Ltd. 80 218 20 Interval Estimate of 1 - 2: Small-Sample Case (n1 < 30 and/or n2 < 30) Interval Estimate with 12 22 2 x1 x2 z / 2 x1 x2 where: x1 x2 1 1 ( ) n1 n2 2 Interval Estimate of 1 - 2: Small-Sample Case (n1 < 30 and/or n2 < 30) • Interval Estimate with 2 Unknown x1 x2 t / 2 sx1 x2 where: sx1 x2 1 1 s ( ) n1 n2 2 2 2 ( n 1 ) s ( n 1 ) s 1 2 2 s2 1 n1 n2 2 Difference Between Two Population Means: Small Sample Case Example: Specific Motors Specific Motors of Detroit has developed a new automobile known as the M car. 12 M cars and 8 J cars (from Japan) were road tested to compare miles-per-gallon (mpg) performance. The sample statistics are shown on the next slide. Difference Between Two Population Means: Small Sample Case Example: Specific Motors Sample #1 M Cars 12 cars 29.8 mpg 2.56 mpg Sample #2 J Cars 8 cars 27.3 mpg 1.81 mpg Sample Size Sample Mean Sample Std. Dev. Point Estimate of the Difference Between Two Population Means Point estimate of 1 2 = x1 x2 = 29.8 - 27.3 = 2.5 mpg where: 1 = mean miles-per-gallon for the population of M cars 2 = mean miles-per-gallon for the population of J cars 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case We will make the following assumptions: • The miles per gallon rating is normally distributed for both the M car and the J car. • The variance in the miles per gallon rating is the same for both the M car and the J car. 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case We will use a weighted average of the two sample variances as the pooled estimator of 2. 2 2 2 2 ( n 1 ) s ( n 1 ) s 11 ( 2 . 56 ) 7 ( 1 . 81 ) 1 2 2 s2 1 5. 28 n1 n2 2 12 8 2 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case Using the t distribution with n1 + n2 - 2 = 18 degrees of freedom, the appropriate t value is t.025 = 2.101. x1 x2 t.025 1 1 1 1 s ( ) 2. 5 2.101 5. 28( ) n1 n2 12 8 2 2.5 + 2.2 or .3 to 4.7 miles per gallon We are 95% confident that the difference between the mean mpg ratings of the two car types is .3 to 4.7 mpg (with the M car having the higher mpg). Using Excel to Develop an Interval Estimate of 1 – 2: Small-Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Formula Worksheet A M Car 25.1 32.2 31.7 27.6 28.5 33.6 30.8 26.2 29.0 31.0 31.7 30.0 B C D J Car M Car 25.6 Sample Size 12 28.1 Mean =AVERAGE(A2:A13) 27.9 Stand. Dev. =STDEV(A2:A13) 25.3 30.1 Confid. Coeff. 0.95 27.5 Lev. of Signif. =1-D6 25.1 Deg. Freed. =D2+E2-2 28.8 z Value =TINV(D7,D8) Pool.Est.Var. =((D2-1)*D4^2+(E2-1)*E4^2)/D8 Std. Error =SQRT(D11*(1/D2+1/E2)) Marg. of Error =D9*D12 Pt. Est. of Diff. =D3-E3 Lower Limit =D15-D13 Upper Limit =D15+D13 E J Car 8 =AVERAGE(B2:B9) =STDEV(B2:B9) Using Excel to Develop an Interval Estimate of 1 – 2: Small-Sample Value Worksheet 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 A M Car 25.1 32.2 31.7 27.6 28.5 33.6 30.8 26.2 29.0 31.0 31.7 30.0 B C J Car 25.6 Sample Size 12 28.1 Mean 29.8 27.9 Stand. Dev. 2.56 25.3 30.1 Confid. Coeff. 0.95 27.5 Lev. of Signif. 0.05 25.1 Deg. Freed. 18 28.8 z Value 2.101 Pool.Est.Var. 5.2765 Std. Error 1.0485 Marg. of Error 2.2027 Pt. Est. of Diff. 2.4833 Lower Limit 0.2806 Upper Limit 4.6861 D M Car E J Car 8 27.3 1.81 Hypothesis Tests About the Difference between the Means of Two Populations: Independent Samples • Hypotheses H 0 : 1 2 0 H a : 1 2 0 H 0 : 1 2 0 H 0 : 1 2 0 H a : 1 2 0 H a : 1 2 0 • Test Statistic Large-Sample z ( x1 x2 ) ( 1 2 ) 12 n1 22 n2 Small-Sample t ( x1 x2 ) ( 1 2 ) s2 (1 n1 1 n2 ) Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Large-Sample Case Example: Par, Inc. Recall that Par, Inc. has developed a new golf ball that was designed to provide “extra distance.” A sample of Par golf balls was compared with a sample of golf balls made by Rap, Ltd., a competitor. The sample statistics appear on the next slide. Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Large-Sample Case Example: Par, Inc. Can we conclude, using = .01, that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls? Sample #1 Sample Size Sample Mean Sample Std. Dev. Par, Inc. 120 balls 235 yards 15 yards Sample #2 Rap, Ltd. 80 balls 218 yards 20 yards Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Large-Sample Case Using the Test Statistic 1. Determine the hypotheses. H0: 1 - 2 < 0 Ha: 1 - 2 > 0 where: 1 = mean distance for the population of Par, Inc. golf balls 2 = mean distance for the population of Rap, Ltd. golf balls Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Large-Sample Case Using the Test Statistic 2. Specify the level of significance. 3. Select the test statistic. z = .01 ( x1 x 2 ) ( 1 2 ) 12 n1 4. State the rejection rule. 22 n2 Reject H0 if z > 2.33 Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Large-Sample Case Using the Test Statistic 5. Compute the value of the test statistic. z ( x1 x 2 ) ( 1 2 ) 12 n1 z 22 n2 (235 218) 0 (15)2 (20)2 120 80 17 6.49 2.62 Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Large-Sample Case Using the Test Statistic 6. Determine whether to reject H0. z = 6.49 > z.01 = 2.33, so we reject H0. At the .01 level of significance, the sample evidence indicates the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls. Using Excel to Conduct a Hypothesis Test about 1 – 2: Large Sample Case Excel’s “z-Test: Two Sample for Means” Tool Step 1 Select the Tools menu Step 2 Choose the Data Analysis option Step 3 Choose z-Test: Two Sample for Means from the list of Analysis Tools … continued Using Excel to Conduct a Hypothesis Test about 1 – 2: Large Sample Case Excel’s “z-Test: Two Sample for Means” Tool Step 4 When the z-Test: Two Sample for Means dialog box appears: Enter A1:A121 in the Variable 1 Range box Enter B1:B81 in the Variable 2 Range box Type 0 in the Hypothesized Mean Difference box Type 225 in the Variable 1 Variance (known) box Type 400 in the Variable 2 Variance (known) box … continued Using Excel to Conduct a Hypothesis Test about 1 – 2: Large Sample Case Excel’s “z-Test: Two Sample for Means” Tool Step 4 (continued) Select Labels Type .01 in the Alpha box Select Output Range Enter D4 in the Output Range box (Any upper left-hand corner cell indicating where the output is to begin may be entered) Click OK Using Excel to Conduct a Hypothesis Test about 1 – 2: Large Sample Case Using Excel to Conduct a Hypothesis Test about 1 – 2: Large Sample Case 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Value Worksheet A Par 195 230 254 205 260 222 241 217 228 255 209 251 229 220 B C D Rap 226 Sample Variance 198 203 z-Test: Two Sample for Means 237 235 204 Mean 199 Known Variance 202 Observations 240 Hypothesized Mean Difference 221 z 206 P(Z<=z) one-tail 201 z Critical one-tail 233 P(Z<=z) two-tail 194 z Critical two-tail Note: Rows 16-121 are not shown. E Par, Inc. 225 F Rap, Ltd. 400 Par, Inc. Rap, Ltd. 235 218 225 400 120 80 0 6.483545607 4.50145E-11 2.326341928 9.00291E-11 2.575834515 Using Excel to Conduct a Hypothesis Test about 1 – 2: Large Sample Case Using the p Value 4. Compute the value of the test statistic. The Excel worksheet states z = 6.48 5. Compute the p–value. The Excel worksheet states p-value = 4.501E-11 6. Determine whether to reject H0. Because p–value < = .01, we reject H0.