#### Transcript INFERENCE: TWO POPULATIONS

統計學 Spring 2004 授課教師：統計系余清祥 日期：2004年3月2日 第三週：比較期望值 Slide 1 Chapter 10 Comparisons Involving Means 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 Inferences about the Difference between the Means of Two Populations: Matched Samples Inferences about the Difference between the Proportions of Two Populations: Slide 2 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 x1 x2 Interval Estimate of Large-Sample Case Interval Estimate of Small-Sample Case Slide 3 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. x2 Let x1 equal the mean of sample 1 and equal the mean of sample 2. The point estimator of the difference between the x1 1 and x2 2 is means of the populations . Slide 4 Sampling Distribution ofx1 x2 Properties of the Sampling Distribution xof 1 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 Slide 5 Interval Estimate of 1 - 2: Large-Sample Case (n1 > 30 and n2 > 30) Interval Estimate with 1 and 2 Known where: x1 x2 z / 2 x1 x2 1 - is the confidence coefficient Interval Estimate with 1 and 2 Unknown x1 x2 z / 2 sx1 x2 where: sx1 x2 s12 s22 n1 n2 Slide 6 Example: Par, Inc. Interval Estimate of 1 - 2: Large-Sample Case 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. Slide 7 Example: Par, Inc. Interval Estimate of 1 - 2: Large-Sample Case • Sample Statistics Sample Size Mean Standard Dev. Sample #1 Par, Inc. n1 = 120 balls x1 = 235 yards s1 = 15 yards Sample #2 Rap, Ltd. n2 = 80 balls x2 = 218 yards s2 = 20 yards Slide 8 Example: Par, Inc. Point Estimate of the Difference Between Two Population Means 1 = mean distance for the population of Par, Inc. golf balls 2 = mean distance for the population of Rap, Ltd. golf balls Point estimate of 1 - 2 =x1 x2 = 235 - 218 = 17 yards. Slide 9 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 Slide 10 Example: Par, Inc. 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 lies in the interval of 11.86 to 22.14 yards. Slide 11 Interval Estimate of 1 - 2: Small-Sample Case (n1 < 30 and/or n2 < 30) Interval Estimate with 2 Known x1 x2 z / 2 x1 x2 where: x1 x2 1 1 ( ) n1 n2 2 Slide 12 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 Slide 13 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-pergallon (mpg) performance. The sample statistics are: Sample Size Mean Standard Deviation Sample #1 M Cars n1 = 12 cars x1 = 29.8 mpg s1 = 2.56 mpg Sample #2 J Cars n2 = 8 cars x2 = 27.3 mpg s2 = 1.81 mpg Slide 14 Example: Specific Motors Point Estimate of the Difference Between Two Population Means 1 = mean miles-per-gallon for the population of M cars 2 = mean miles-per-gallon for the population of J cars Point estimate of 1 - 2 =x1 x2 = 29.8 - 27.3 = 2.5 mpg. Slide 15 Example: Specific Motors 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 must be normally distributed for both the M car and the J car. • The variance in the miles per gallon rating must be the same for both the M car and the J car. Using the t distribution with n1 + n2 - 2 = 18 degrees of freedom, the appropriate t value is t.025 = 2.101. We will use a weighted average of the two sample variances as the pooled estimator of 2. Slide 16 Example: Specific Motors 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case 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 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 from .3 to 4.7 mpg (with the M car having the higher mpg). Slide 17 Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples Hypotheses H0: 1 - 2 < 0 Ha: 1 - 2 > 0 H0: 1 - 2 > 0 Ha: 1 - 2 < 0 Test Statistic Large-Sample z ( x1 x2 ) ( 1 2 ) 12 n1 22 n2 H0: 1 - 2 = 0 Ha: 1 - 2 0 Small-Sample t ( x1 x2 ) ( 1 2 ) s2 (1 n1 1 n2 ) Slide 18 Example: Par, Inc. Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case 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. Slide 19 Example: Par, Inc. Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case • Sample Statistics Sample Size Mean Standard Dev. Sample #1 Par, Inc. n1 = 120 balls x1 = 235 yards s1 = 15 yards Sample #2 Rap, Ltd. n2 = 80 balls x2 = 218 yards s2 = 20 yards Slide 20 Example: Par, Inc. Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case Can we conclude, using a .01 level of significance, that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls? 1 = mean distance for the population of Par, Inc. golf balls 2 = mean distance for the population of Rap, Ltd. golf balls • Hypotheses H0: 1 - 2 < 0 Ha: 1 - 2 > 0 Slide 21 Example: Par, Inc. Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case • Rejection Rule Reject H0 if z > 2.33 z ( x1 x2 ) ( 1 2 ) 12 n1 22 n2 ( 235 218) 0 17 6. 49 2 2 2. 62 (15) ( 20) 120 80 • Conclusion Reject H0. We are at least 99% confident that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls. Slide 22 Example: Specific Motors Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case Can we conclude, using a .05 level of significance, that the miles-per-gallon (mpg) performance of M cars is greater than the miles-pergallon performance of J cars? 1 = mean mpg for the population of M cars 2 = mean mpg for the population of J cars • Hypotheses H0: 1 - 2 < 0 Ha: 1 - 2 > 0 Slide 23 Example: Specific Motors Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case • Rejection Rule Reject H0 if t > 1.734 (a = .05, d.f. = 18) • Test Statistic t where: ( x1 x2 ) ( 1 2 ) s2 (1 n1 1 n2 ) (n1 1)s12 (n2 1)s22 s n1 n2 2 2 Slide 24 Inference About the Difference Between the Means of Two Populations: Matched Samples With a matched-sample design each sampled item provides a pair of data values. The matched-sample design can be referred to as blocking. This design often leads to a smaller sampling error than the independent-sample design because variation between sampled items is eliminated as a source of sampling error. Slide 25 Example: Express Deliveries Inference About the Difference Between the Means of Two Populations: Matched Samples A Chicago-based firm has documents that must be quickly distributed to district offices throughout the U.S. The firm must decide between two delivery services, UPX (United Parcel Express) and INTEX (International Express), to transport its documents. In testing the delivery times of the two services, the firm sent two reports to a random sample of ten district offices with one report carried by UPX and the other report carried by INTEX. Do the data that follow indicate a difference in mean delivery times for the two services? Slide 26 Example: Express Deliveries District Office Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee Denver Delivery Time (Hours) UPX INTEX Difference 32 30 19 16 15 18 14 10 7 16 25 24 15 15 13 15 15 8 9 11 7 6 4 1 2 3 -1 2 -2 5 Slide 27 Example: Express Deliveries Inference About the Difference Between the Means of Two Populations: Matched Samples Let d = the mean of the difference values for the two delivery services for the population of district offices • Hypotheses • Rejection Rule H0: d = 0, Ha: d Assuming the population of difference values is approximately normally distributed, the t distribution with n - 1 degrees of freedom applies. With = .05, t.025 = 2.262 (9 degrees of freedom). Reject H0 if t < -2.262 or if t > 2.262 Slide 28 Example: Express Deliveries Inference About the Difference Between the Means of Two Populations: Matched Samples di ( 7 6... 5) d 2. 7 n 10 2 76.1 ( di d ) sd 2. 9 n 1 9 d d 2. 7 0 t 2. 94 sd n 2. 9 10 • Conclusion Reject H0. There is a significant difference between the mean delivery times for the two services. Slide 29 Inferences About the Difference Between the Proportions of Two Populations Sampling Distribution of p1 p2 Interval Estimation of p1 - p2 Hypothesis Tests about p1 - p2 Slide 30 Sampling Distribution of p1 p2 Expected Value E ( p1 p2 ) p1 p2 Standard Deviation p1 p2 p1 (1 p1 ) p2 (1 p2 ) n1 n2 Distribution Form If the sample sizes are large (n1p1, n1(1 - p1), n2p2, and n2(1 - p2) are all greater than or equal to 5), the sampling distribution of p1 p2 can be approximated by a normal probability distribution. Slide 31 Interval Estimation of p1 - p2 Interval Estimate p1 p2 z / 2 p1 p2 Point Estimator of p1 p2 s p1 p2 p1 (1 p1 ) p2 (1 p2 ) n1 n2 Slide 32 Example: MRA MRA (Market Research Associates) is conducting research to evaluate the effectiveness of a client’s new advertising campaign. Before the new campaign began, a telephone survey of 150 households in the test market area showed 60 households “aware” of the client’s product. The new campaign has been initiated with TV and newspaper advertisements running for three weeks. A survey conducted immediately after the new campaign showed 120 of 250 households “aware” of the client’s product. Does the data support the position that the advertising campaign has provided an increased awareness of the client’s product? Slide 33 Example: MRA Point Estimator of the Difference Between the Proportions of Two Populations 120 60 p1 p2 p1 p2 . 48. 40 . 08 250 150 p1 = proportion of the population of households “aware” of the product after the new campaign p2 = proportion of the population of households “aware” of the product before the new p1 campaign = sample proportion of households “aware” of the p2 product after the new campaign = sample proportion of households “aware” of the product before the new campaign Slide 34 Example: MRA Interval Estimate of p1 - p2: Large-Sample Case For = .05, z.025 = 1.96: . 48(.52) . 40(. 60) . 48. 40 1. 96 250 150 .08 + 1.96(.0510) .08 + .10 or -.02 to +.18 • Conclusion At a 95% confidence level, the interval estimate of the difference between the proportion of households aware of the client’s product before and after the new advertising campaign is -.02 to +.18. Slide 35 Hypothesis Tests about p1 - p2 Hypotheses H0: p1 - p2 < 0 Ha: p1 - p2 > 0 Test statistic z ( p1 p2 ) ( p1 p2 ) p1 p2 Point Estimator of p1 p2 where p1 = p2 s p1 p2 p (1 p )(1 n1 1 n2 ) where: n1 p1 n2 p2 p n1 n2 Slide 36 Example: MRA Hypothesis Tests about p1 - p2 Can we conclude, using a .05 level of significance, that the proportion of households aware of the client’s product increased after the new advertising campaign? p1 = proportion of the population of households “aware” of the product after the new campaign p2 = proportion of the population of households “aware” of the product before the new campaign • Hypotheses H0: p1 - p2 < 0 Ha: p1 - p2 > 0 Slide 37 Example: MRA Hypothesis Tests about p1 - p2 • Rejection Rule Reject H0 if z > 1.645 • Test Statistic 250(. 48) 150(. 40) 180 p . 45 250 150 400 s p1 p2 . 45(.55)( 1 1 ) . 0514 250 150 (. 48. 40) 0 . 08 z 1.56 . 0514 . 0514 • Conclusion Do not reject H0. Slide 38 End of Chapter 10 Slide 39