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Statistical Inference: Estimating the Difference Between Two Population Means

Learn techniques for estimating the difference between two population means using sample statistics, confidence intervals, and hypothesis testing. Understand the process with a practical example comparing golf ball driving distances.

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Statistical Inference: Estimating the Difference Between Two Population Means

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  1. Slides by John Loucks St. Edward’s University

  2. Chapter 10, Part A Statistical Inference About Means and Proportions With Two Populations • Inferences About the Difference Between Two Population Means: s1 and s2 Known • Inferences About the Difference Between Two Population Means: s1 and s2 Unknown

  3. Let equal the mean of sample 1 and equal the mean of sample 2. • The point estimator of the difference between the • means of the populations 1 and 2 is . Estimating the Difference BetweenTwo 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.

  4. Sampling Distribution of • Expected Value • Standard Deviation (Standard Error) 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

  5. Interval Estimation of 1 - 2:s 1 and s 2 Known • Interval Estimate where: 1 -  is the confidence coefficient

  6. 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.

  7. Interval Estimation of 1 - 2:s 1 and s 2 Known • Example: Par, Inc. Sample #1 Par, Inc. Sample #2 Rap, Ltd. Sample Size 120 balls 80 balls Sample Mean 275 yards 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.

  8. 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.

  9. Population 1 Par, Inc. Golf Balls m1 = mean driving distance of Par golf balls Population 2 Rap, Ltd. Golf Balls m2 = mean driving distance of Rap golf balls Simple random sample of n1 Par golf balls x1 = sample mean distance for the Par golf balls Simple random sample of n2 Rap golf balls x2 = sample mean distance for the Rap golf balls x1 - x2 = Point Estimate of m1 –m2 Estimating the Difference BetweenTwo Population Means m1 –m2= difference between the mean distances

  10. Point Estimate of 1 - 2 Point estimate of 1-2 = = 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

  11. Interval Estimation of 1 - 2:1 and 2 Known 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.

  12. A B C D E 1 Par Rap Par, Inc. Rap, Ltd. 2 255 266 Sample Size =COUNT(A2:A121) =COUNT(B2:B81) 3 270 238 Sample Mean =AVERAGE(A2:A121) =AVERAGE(B2:B81) 4 294 243 5 245 277 Popul. Std. Dev. 15 20 6 300 275 Standard Error =SQRT(D5^2/D2+E5^2/E2) 7 262 244 8 281 239 Confid. Coeff. 0.95 9 257 242 Level of Signif. =1-D8 10 268 280 z Value =NORM.S.INV(1-D9/2) 11 295 261 Margin of Error =D10*D6 12 249 276 13 291 241 Pt. Est. of Diff. =D3-E3 14 289 273 Lower Limit =D13-D11 15 282 248 Upper Limit =D13+D11 Interval Estimation of 1 - 2:1 and 2 Known • Excel Formula Worksheet Note: Rows 16-121 are not shown.

  13. Interval Estimation of 1 - 2:1 and 2 Known • Excel Value Worksheet Note: Rows 16-121 are not shown.

  14. Hypothesis Tests About m 1-m 2:s 1 and s 2 Known • Hypotheses Left-tailed Right-tailed Two-tailed • Test Statistic

  15. Hypothesis Tests About m 1-m 2:s 1 and s 2 Known • Example: Par, Inc. Can we conclude, using a = .01, that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. Golf balls?

  16. Hypothesis Tests About m 1-m 2:s 1 and s 2 Known • p –Value and Critical Value Approaches Right- tailed test 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 a = .01 2. Specify the level of significance.

  17. Hypothesis Tests About m 1-m 2:s 1 and s 2 Known • p –Value and Critical Value Approaches 3. Compute the value of the test statistic.

  18. Hypothesis Tests About m 1-m 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 <a = .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.

  19. Hypothesis Tests About m 1-m 2:s 1 and s 2 Known • Critical Value Approach 4. Determine the critical value and rejection rule. For a = .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.

  20. Excel’s “z-Test: Two Sample for Means” Tool Step 1Click the Data tab on the Ribbon Step 2In 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)

  21. Excel’s “z-Test: Two Sample for Means” Tool • Excel Dialog Box

  22. Excel’s “z-Test: Two Sample for Means” Tool • Excel Value Worksheet Note: Rows 14-121 are not shown.

  23. 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 za/2 with ta/2.

  24. Interval Estimation of 1 - 2:s 1 and s 2 Unknown • Interval Estimate Where the degrees of freedom for ta/2 are:

  25. 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.

  26. Difference Between Two Population Means: s 1 and s 2 Unknown • Example: Specific Motors Sample #1 M Cars Sample #2 J Cars Sample Size 24 cars 28 cars Sample Mean 29.8 mpg 27.3 mpg Sample Std. Dev. 2.56 mpg 1.81 mpg

  27. 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.

  28. Point Estimate of m 1-m 2 Point estimate of 1-2 = = 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

  29. Interval Estimation of m 1-m 2:s 1 and s 2 Unknown The degrees of freedom for ta/2 are: With a/2 = .05 and df = 24, ta/2 = 1.711

  30. Interval Estimation of m 1-m 2:s 1 and s 2 Unknown 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.

  31. A B C D E 1 M J Par, Inc. Rap, Ltd. 2 26.1 25.6 Sample Size =COUNT(A2:A25) =COUNT(B2:B29) 3 32.5 28.1 Sample Mean =AVERAGE(A2:A25) =AVERAGE(B2:B29) 4 31.8 27.9 Sample Std. Dev. =STDEV(A2:A25) =STDEV(B2:B29) 5 27.6 25.3 6 28.5 30.1 Est. of Variance =D4^2/D2+E4^2/E2 7 33.6 27.5 Standard Error =SQRT(D6) 8 31.7 26.0 9 25.2 28.8 Confid. Coeff. 0.90 10 26.0 30.6 Level of Signif. =1-D9 11 32.0 24.4 Degr. of Freedom =D6^2/((1/(D2-1))*(D4^2/D2)^2+(1/(E2-1))*(E4^2/E2)^2)) 12 31.7 27.3 t Value =T.INV.2T(D10,D11) 13 30.4 27.5 Margin of Error =D12*D7 14 27.6 26.3 15 32.3 25.5 Point Est. of Diff. =D3-E3 16 30.6 26.3 Lower Limit =D15-D13 17 29.5 24.3 Upper Limit =D15+D13 Interval Estimation of m 1-m 2:s 1 and s 2 Unknown • Excel Formula Worksheet Note: Rows 18-29 are not shown.

  32. Interval Estimation of m 1-m 2:s 1 and s 2 Unknown • Excel Formula Worksheet Note: Rows 18-29 are not shown.

  33. Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown • Hypotheses Left-tailed Right-tailed Two-tailed • Test Statistic

  34. Hypothesis Tests About m 1-m 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?

  35. Hypothesis Tests About m 1-m 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 Right- tailed test where: 1 = mean mpg for the population of M cars 2 = mean mpg for the population of J cars

  36. Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown • p –Value and Critical Value Approaches a = .05 2. Specify the level of significance. 3. Compute the value of the test statistic.

  37. Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown • p –Value Approach 4. Compute the p –value. The degrees of freedom for ta are: Because t = 4.003 > t.005 = 1.683, the p–value < .005.

  38. Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown • p –Value Approach 5. Determine whether to reject H0. Because p–value <a = .05, we reject H0. We are at least 95% confident that the miles-per-gallon (mpg) performance of M cars is greater than the miles-per-gallon performance of J cars.

  39. Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown • Critical Value Approach 4. Determine the critical value and rejection rule. For a = .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-per-gallon (mpg) performance of M cars is greater than the miles-per-gallon performance of J cars.

  40. Excel’s “z-Test: Two-SampleAssuming Unequal Variances” Tool Step 1Click the Data tab on the Ribbon Step 2In 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)

  41. Excel’s “z-Test: Two-SampleAssuming Unequal Variances” Tool • Excel Dialog Box

  42. Excel’s “z-Test: Two-SampleAssuming Unequal Variances” Tool • Excel Value Worksheet Note: Rows 14-121 are not shown.

  43. End of Chapter 10, Part A

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