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Slides by John Loucks St. Edward’s University © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 1 Chapter 10, Part A Statistical Inference About Means and Proportions With Two Populations Inferences About the Difference Between Two Population Means: s 1 and s 2 Known Inferences About the Difference Between Two Population Means: s 1 and s 2 Unknown © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 2 Estimating the Difference Between Two Population Means 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. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 3 Sampling Distribution of x1 x2 Expected Value E ( x1 x2 ) 1 2 Standard Deviation (Standard Error) s x1 x2 s12 n1 s 22 n2 where: s1 = standard deviation of population 1 s2 = standard deviation of population 2 n1 = sample size from population 1 n2 = sample size from population 2 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 4 Interval Estimation of 1 - 2: s 1 and s 2 Known Interval Estimate x1 x2 z / 2 s 12 s 22 n1 n2 where: 1 - is the confidence coefficient © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 5 Interval Estimation of 1 - 2: s 1 and s 2 Known 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. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 6 Interval Estimation of 1 - 2: s 1 and s 2 Known Example: Par, Inc. Sample Size Sample Mean Sample #1 Par, Inc. 120 balls 275 yards Sample #2 Rap, Ltd. 80 balls 258 yards Based on data from previous driving distance tests, the two population standard deviations are known with s 1 = 15 yards and s 2 = 20 yards. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 7 Interval Estimation of 1 - 2: s 1 and s 2 Known Example: Par, Inc. Let us develop a 95% confidence interval estimate of the difference between the mean driving distances of the two brands of golf ball. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 8 Estimating the Difference Between Two Population Means Population 1 Par, Inc. Golf Balls 1 = mean driving distance of Par golf balls Population 2 Rap, Ltd. Golf Balls 2 = mean driving 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 the Par golf balls x2 = sample mean distance for the Rap golf balls x1 - x2 = Point Estimate of m1 – 2 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 9 Point Estimate of 1 - 2 Point estimate of 1 2 = x1 x2 = 275 258 = 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 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 10 Interval Estimation of 1 - 2: s 1 and s 2 Known x1 x2 z / 2 s12 s 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. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 11 Interval Estimation of 1 - 2: s 1 and s 2 Known 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Excel Formula Worksheet A Par 255 270 294 245 300 262 281 257 268 295 249 291 289 282 B C D E Rap Par, Inc. Rap, Ltd. 266 Sample Size =COUNT(A2:A121) =COUNT(B2:B81) 238 Sample Mean =AVERAGE(A2:A121) =AVERAGE(B2:B81) 243 277 Popul. Std. Dev. 15 20 275 Standard Error =SQRT(D5^2/D2+E5^2/E2) 244 239 Confid. Coeff. 0.95 242 Level of Signif. =1-D8 280 z Value =NORM.S.INV(1-D9/2) 261 Margin of Error =D10*D6 276 241 Pt. Est. of Diff. =D3-E3 273 Lower Limit =D13-D11 248 Upper Limit =D13+D11 Note: Rows 16-121 are not shown. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 12 Interval Estimation of 1 - 2: s 1 and s 2 Known 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Excel Value Worksheet A Par 255 270 294 245 300 262 281 257 268 295 249 291 289 282 C B Rap Sample Size 120 266 Sample Mean 275 238 243 277 Popul. Std. Dev. 15 275 Standard Error 2.622 244 Confid. Coeff. 0.95 239 242 Level of Signif. 0.05 z Value 1.96 280 261 Margin of Error 5.14 276 Pt. Est. of Diff. 17 241 Lower Limit 11.86 273 Upper Limit 22.14 248 E Rap, Ltd. D Par, Inc. 80 258 20 Note: Rows 16-121 are not shown. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 13 Hypothesis Tests About 1 2: s 1 and s 2 Known Hypotheses H0 : 1 2 D0 H0 : 1 2 D0 H0 : 1 2 D0 H a : 1 2 D0 H a : 1 2 D0 H a : 1 2 D0 Left-tailed Right-tailed Two-tailed Test Statistic z ( x1 x2 ) D0 s 12 n1 s 22 n2 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 14 Hypothesis Tests About 1 2: s 1 and s 2 Known 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? © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 15 Hypothesis Tests About 1 2: s 1 and s 2 Known p –Value and Critical Value Approaches 1. Develop 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 2. Specify the level of significance. Righttailed test = .01 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 16 Hypothesis Tests About 1 2: s 1 and s 2 Known p –Value and Critical Value Approaches 3. Compute the value of the test statistic. z ( x1 x2 ) D0 s 12 n1 z s 22 n2 (235 218) 0 (15)2 (20)2 120 80 17 6.49 2.62 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 17 Hypothesis Tests About 1 2: s 1 and s 2 Known p –Value Approach 4. Compute the p–value. For z = 6.49, the p –value < .0001. 5. Determine whether to reject H0. Because p–value < = .01, 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. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 18 Hypothesis Tests About 1 2: s 1 and s 2 Known Critical Value Approach 4. Determine the critical value and rejection rule. For = .01, z.01 = 2.33 Reject H0 if z > 2.33 5. Determine whether to reject H0. Because z = 6.49 > 2.33, we reject H0. 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. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 19 Excel’s “z-Test: Two Sample for Means” Tool Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose z-Test: Two Sample for Means from the list of Analysis Tools Step 4 When the z-Test: Two Sample for Means dialog box appears: (see details on next slide) © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 20 Excel’s “z-Test: Two Sample for Means” Tool Excel Dialog Box © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 21 Excel’s “z-Test: Two Sample for Means” Tool 1 2 3 4 5 6 7 8 9 10 11 12 13 Excel Value Worksheet A Par 255 270 294 245 300 262 281 257 268 295 249 291 Note: B C D Rap 266 z-Test: Two Sample for Means 238 243 277 Mean 275 Known Variance 244 Observations 239 Hypothesized Mean Difference 242 z 280 P(Z<=z) one-tail 261 z Critical one-tail 276 P(Z<=z) two-tail 241 z Critical two-tail Rows 14-121 are not shown. E F Par, Inc. Rap, Ltd. 235 218 225 400 120 80 0 6.483545607 4.50145E-11 2.326341928 9.00291E-11 2.575834515 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 22 Interval Estimation of 1 - 2: s 1 and s 2 Unknown When s 1 and s 2 are unknown, we will: • use the sample standard deviations s1 and s2 as estimates of s 1 and s 2 , and • replace z/2 with t/2. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 23 Interval Estimation of 1 - 2: s 1 and s 2 Unknown Interval Estimate x1 x2 t / 2 s12 s22 n1 n2 Where the degrees of freedom for t/2 are: 2 s s n1 n2 df 2 2 2 2 1 s1 1 s2 n1 1 n1 n2 1 n2 2 1 2 2 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 24 Difference Between Two Population Means: s 1 and s 2 Unknown Example: Specific Motors Specific Motors of Detroit has developed a new automobile known as the M car. 24 M cars and 28 J cars (from Japan) were road tested to compare miles-per-gallon (mpg) performance. The sample statistics are shown on the next slide. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 25 Difference Between Two Population Means: s 1 and s 2 Unknown Example: Specific Motors Sample #1 M Cars 24 cars Sample Size Sample Mean 29.8 mpg Sample Std. Dev. 2.56 mpg Sample #2 J Cars 28 cars 27.3 mpg 1.81 mpg © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 26 Difference Between Two Population Means: s 1 and s 2 Unknown Example: Specific Motors Let us develop a 90% confidence interval estimate of the difference between the mpg performances of the two models of automobile. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 27 Point Estimate of 1 2 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 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 28 Interval Estimation of 1 2: s 1 and s 2 Unknown The degrees of freedom for t/2 are: 2 (2.56) (1.81) 24 28 df 24.07 24 2 2 1 (2.56) 2 1 (1.81) 2 24 1 24 28 1 28 2 2 With /2 = .05 and df = 24, t/2 = 1.711 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 29 Interval Estimation of 1 2: s 1 and s 2 Unknown x1 x2 t / 2 s12 s22 (2.56)2 (1.81)2 29.8 27.3 1.711 n1 n2 24 28 2.5 + 1.069 or 1.431 to 3.569 mpg We are 90% confident that the difference between the miles-per-gallon performances of M cars and J cars is 1.431 to 3.569 mpg. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 30 Interval Estimation of 1 2: s 1 and s 2 Unknown 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Excel Formula Worksheet A M 26.1 32.5 31.8 27.6 28.5 33.6 31.7 25.2 26.0 32.0 31.7 30.4 27.6 32.3 30.6 29.5 B C D E J Par, Inc. Rap, Ltd. 25.6 Sample Size =COUNT(A2:A25) =COUNT(B2:B29) 28.1 Sample Mean =AVERAGE(A2:A25) =AVERAGE(B2:B29) 27.9 Sample Std. Dev. =STDEV(A2:A25) =STDEV(B2:B29) 25.3 30.1 Est. of Variance =D4^2/D2+E4^2/E2 27.5 Standard Error =SQRT(D6) 26.0 28.8 Confid. Coeff. 0.90 30.6 Level of Signif. =1-D9 24.4 Degr. of Freedom =D6^2/((1/(D2-1))*(D4^2/D2)^2+(1/(E2-1))*(E4^2/E2)^2)) 27.3 t Value =T.INV.2T(D10,D11) 27.5 Margin of Error =D12*D7 26.3 Note: 25.5 Point Est. of Diff.=D3-E3 Rows 18-29 are not shown. 26.3 Lower Limit =D15-D13 24.3 Upper Limit =D15+D13 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 31 Interval Estimation of 1 2: s 1 and s 2 Unknown Excel Formula Worksheet A B C D E 1 M J Par, Inc. Rap, Ltd. 2 26.1 25.6 Sample Size 24 28 3 32.5 28.1 Sample Mean 29.8 27.3 4 31.8 27.9 Sample Std. Dev. 2.56 1.81 5 27.6 25.3 6 28.5 30.1 Est. of Variance 0.39007 7 33.6 27.5 Standard Error 0.62456 8 31.7 26.0 9 25.2 28.8 Confid. Coeff. 0.90 10 26.0 30.6 Level of Signif. 0.10 11 32.0 24.4 Degr. of Freedom 24.07 12 31.7 27.3 t Value 1.711 13 30.4 27.5 Margin of Error 1.069 14 27.6 26.3 Note: 15 32.3 25.5 Point Est. of Diff. 2.5 Rows 18-29 16 30.6 26.3 Lower Limit 1.431 are not shown. 17 29.5 24.3 Upper Limit 3.569 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied Slide 32 or duplicated, or posted to a publicly accessible website, in whole or in part. Hypothesis Tests About 1 2: s 1 and s 2 Unknown Hypotheses H0 : 1 2 D0 H0 : 1 2 D0 H0 : 1 2 D0 H a : 1 2 D0 H a : 1 2 D0 H a : 1 2 D0 Left-tailed Right-tailed Two-tailed Test Statistic t ( x1 x2 ) D0 2 1 2 2 s s n1 n2 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 33 Hypothesis Tests About 1 2: s 1 and s 2 Unknown Example: Specific Motors Can we conclude, using a .05 level of significance, that the miles-per-gallon (mpg) performance of M cars is greater than the miles-per-gallon performance of J cars? © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 34 Hypothesis Tests About 1 2: s 1 and s 2 Unknown p –Value and Critical Value Approaches 1. Develop the hypotheses. H0: 1 - 2 < 0 Ha: 1 - 2 > 0 Righttailed test where: 1 = mean mpg for the population of M cars 2 = mean mpg for the population of J cars © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 35 Hypothesis Tests About 1 2: s 1 and s 2 Unknown p –Value and Critical Value Approaches 2. Specify the level of significance. = .05 3. Compute the value of the test statistic. t ( x1 x2 ) D0 s12 s22 n1 n2 (29.8 27.3) 0 (2.56)2 (1.81)2 24 28 4.003 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 36 Hypothesis Tests About 1 2: s 1 and s 2 Unknown p –Value Approach 4. Compute the p –value. The degrees of freedom for t are: 2 (2.56) (1.81) 24 28 df 40.566 41 2 2 1 (2.56) 2 1 (1.81) 2 24 1 24 28 1 28 2 2 Because t = 4.003 > t.005 = 1.683, the p–value < .005. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 37 Hypothesis Tests About 1 2: s 1 and s 2 Unknown p –Value Approach 5. Determine whether to reject H0. Because p–value < = .05, we reject H0. We are at least 95% confident that the miles-pergallon (mpg) performance of M cars is greater than the miles-per-gallon performance of J cars. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 38 Hypothesis Tests About 1 2: s 1 and s 2 Unknown Critical Value Approach 4. Determine the critical value and rejection rule. For = .05 and df = 41, t.05 = 1.683 Reject H0 if t > 1.683 5. Determine whether to reject H0. Because 4.003 > 1.683, we reject H0. We are at least 95% confident that the miles-pergallon (mpg) performance of M cars is greater than the miles-per-gallon performance of J cars. © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 39 Excel’s “z-Test: Two-Sample Assuming Unequal Variances” Tool Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose t-Test: Two-Sample Assuming Unequal Variances from the list of Analysis Tools Step 4 When the t-Test: Two-Sample Assuming Unequal Variances dialog box appears: (see details on next slide) © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 40 Excel’s “z-Test: Two-Sample Assuming Unequal Variances” Tool Excel Dialog Box © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 41 Excel’s “z-Test: Two-Sample Assuming Unequal Variances” Tool Excel Value Worksheet A 1 2 3 4 5 6 7 8 9 10 11 12 13 B Mcar Jcar 26.1 32.5 31.8 27.6 28.5 33.6 31.7 25.2 26.0 32.0 31.7 30.4 Note: C D E F z-Test: Two-Sample Assuming Unequal Variances 25.6 28.1 27.9 Mean 25.3 Variance 30.1 Observations 27.5 Hypothesized Mean Difference 26.0 df 28.8 t Stat 30.6 P(T<=t) one-tail 24.4 t Critical one-tail 27.3 P(T<=t) two-tail 27.5 t Critical two-tail Rows 14-121 are not shown. Mcar 29.79583 6.555199 24 0 41 3.99082 1.33000E-04 1.682879 0.000266 2.019542 © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Jcar 27.30357 3.272209 28 Slide 42 End of Chapter 10, Part A © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 43