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Chapter 10 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 Inferences About the Difference Between Two Population Means: Matched Samples Inferences About the Difference Between Two Population Proportions © 2005 Thomson/South-Western Slide 1 Inferences About the Difference Between Two Population Means: s 1 and s 2 Known Interval Estimation of m 1 – m 2 Hypothesis Tests About m 1 – m 2 © 2005 Thomson/South-Western Slide 2 Estimating the Difference Between Two Population Means Let m1 equal the mean of population 1 and m2 equal the mean of population 2. The difference between the two population means is m1 - m2. To estimate m1 - m2, 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. © 2005 Thomson/South-Western Slide 3 Sampling Distribution of x1 x2 Expected Value E ( x1 x2 ) m1 m 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 © 2005 Thomson/South-Western Slide 4 Interval Estimation of m1 - m2: s 1 and s 2 Known Interval Estimate x1 x2 z / 2 s 12 s 22 n1 n2 where: 1 - is the confidence coefficient © 2005 Thomson/South-Western Slide 5 Interval Estimation of m1 - m2: 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. © 2005 Thomson/South-Western Slide 6 Interval Estimation of m1 - m2: 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. © 2005 Thomson/South-Western Slide 7 Interval Estimation of m1 - m2: 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. © 2005 Thomson/South-Western Slide 8 Estimating the Difference Between Two Population Means 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 m1 – m2 = 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 – m2 © 2005 Thomson/South-Western Slide 9 Point Estimate of m1 - m2 Point estimate of m1 m2 = x1 x2 = 275 258 = 17 yards where: m1 = mean distance for the population of Par, Inc. golf balls m2 = mean distance for the population of Rap, Ltd. golf balls © 2005 Thomson/South-Western Slide 10 Interval Estimation of m1 - m2: 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. © 2005 Thomson/South-Western Slide 11 Hypothesis Tests About m 1 m 2: s 1 and s 2 Known Hypotheses H0 : m1 m2 D0 H0 : m1 m2 D0 H0 : m1 m2 D0 H a : m1 m2 D0 H a : m1 m2 D0 H a : m1 m2 D0 Left-tailed Right-tailed Two-tailed Test Statistic z ( x1 x2 ) D0 s 12 n1 © 2005 Thomson/South-Western s 22 n2 Slide 12 Hypothesis Tests About m 1 m 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? © 2005 Thomson/South-Western Slide 13 Hypothesis Tests About m 1 m 2: s 1 and s 2 Known p –Value and Critical Value Approaches 1. Develop the hypotheses. H0: m1 - m2 < 0 Ha: m1 - m2 > 0 where: m1 = mean distance for the population of Par, Inc. golf balls m2 = mean distance for the population of Rap, Ltd. golf balls 2. Specify the level of significance. © 2005 Thomson/South-Western = .01 Slide 14 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. z ( x1 x2 ) D0 s 12 n1 z s 22 n2 (235 218) 0 (15)2 (20)2 120 80 © 2005 Thomson/South-Western 17 6.49 2.62 Slide 15 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 < = .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. © 2005 Thomson/South-Western Slide 16 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 = .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. © 2005 Thomson/South-Western Slide 17 Inferences About the Difference Between Two Population Means: s 1 and s 2 Unknown Interval Estimation of m 1 – m 2 Hypothesis Tests About m 1 – m 2 © 2005 Thomson/South-Western Slide 18 Interval Estimation of m1 - m2: 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. © 2005 Thomson/South-Western Slide 19 Interval Estimation of m1 - m2: 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 © 2005 Thomson/South-Western 2 2 Slide 20 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. © 2005 Thomson/South-Western Slide 21 Difference Between Two Population Means: s 1 and s 2 Unknown Example: Specific Motors Sample #1 M Cars 24 cars 29.8 mpg 2.56 mpg Sample #2 J Cars 28 cars 27.3 mpg 1.81 mpg © 2005 Thomson/South-Western Sample Size Sample Mean Sample Std. Dev. Slide 22 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. © 2005 Thomson/South-Western Slide 23 Point Estimate of m 1 m 2 Point estimate of m1 m2 = x1 x2 = 29.8 - 27.3 = 2.5 mpg where: m1 = mean miles-per-gallon for the population of M cars m2 = mean miles-per-gallon for the population of J cars © 2005 Thomson/South-Western Slide 24 Interval Estimation of m 1 m 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 © 2005 Thomson/South-Western Slide 25 Interval Estimation of m 1 m 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. © 2005 Thomson/South-Western Slide 26 Hypothesis Tests About m 1 m 2: s 1 and s 2 Unknown Hypotheses H0 : m1 m2 D0 H0 : m1 m2 D0 H0 : m1 m2 D0 H a : m1 m2 D0 H a : m1 m2 D0 H a : m1 m2 D0 Left-tailed Right-tailed Two-tailed Test Statistic t ( x1 x2 ) D0 2 1 2 2 s s n1 n2 © 2005 Thomson/South-Western Slide 27 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-pergallon performance of J cars? © 2005 Thomson/South-Western Slide 28 Hypothesis Tests About m 1 m 2: s 1 and s 2 Unknown p –Value and Critical Value Approaches 1. Develop the hypotheses. H0: m1 - m2 < 0 Ha: m1 - m2 > 0 where: m1 = mean mpg for the population of M cars m2 = mean mpg for the population of J cars © 2005 Thomson/South-Western Slide 29 Hypothesis Tests About m 1 m 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 © 2005 Thomson/South-Western (29.8 27.3) 0 (2.56)2 (1.81)2 24 28 4.003 Slide 30 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 t are: (2.56) (1.81) 24 28 2 df 2 2 2 1 (2.56)2 1 (1.81)2 24 1 24 28 1 28 2 24.07 24 Because t = 4.003 > t.005 = 2.797, the p–value < .005. © 2005 Thomson/South-Western Slide 31 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 < = .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?. © 2005 Thomson/South-Western Slide 32 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 = .05 and df = 24, t.05 = 1.711 Reject H0 if t > 1.711 5. Determine whether to reject H0. Because 4.003 > 1.711, 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?. © 2005 Thomson/South-Western Slide 33 Inferences About the Difference Between Two Population Means: Matched Samples With a matched-sample design each sampled item provides a pair of data values. 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. © 2005 Thomson/South-Western Slide 34 Inferences About the Difference Between Two Population Means: Matched Samples Example: Express Deliveries 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. © 2005 Thomson/South-Western Slide 35 Inferences About the Difference Between Two Population Means: Matched Samples Example: Express Deliveries In testing the delivery times of the two services, the firm sent two reports to a random sample of its district offices with one report carried by UPX and the other report carried by INTEX. Do the data on the next slide indicate a difference in mean delivery times for the two services? Use a .05 level of significance. © 2005 Thomson/South-Western Slide 36 Inferences About the Difference Between Two Population Means: Matched Samples Delivery Time (Hours) District Office UPX INTEX Difference Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee Denver 32 30 19 16 15 18 14 10 7 16 © 2005 Thomson/South-Western 25 24 15 15 13 15 15 8 9 11 7 6 4 1 2 3 -1 2 -2 5 Slide 37 Inferences About the Difference Between Two Population Means: Matched Samples p –Value and Critical Value Approaches 1. Develop the hypotheses. H0: md = 0 Ha: md Let md = the mean of the difference values for the two delivery services for the population of district offices © 2005 Thomson/South-Western Slide 38 Inferences About the Difference Between Two Population Means: Matched Samples p –Value and Critical Value Approaches 2. Specify the level of significance. = .05 3. Compute the value of the test statistic. di ( 7 6... 5) d 2. 7 n 10 2 76.1 ( di d ) sd 2. 9 n 1 9 d md 2.7 0 t 2.94 sd n 2.9 10 © 2005 Thomson/South-Western Slide 39 Inferences About the Difference Between Two Population Means: Matched Samples p –Value Approach 4. Compute the p –value. For t = 2.94 and df = 9, the p–value is between .02 and .01. (This is a two-tailed test, so we double the upper-tail areas of .01 and .005.) 5. Determine whether to reject H0. Because p–value < = .05, we reject H0. We are at least 95% confident that there is a difference in mean delivery times for the two services? © 2005 Thomson/South-Western Slide 40 Inferences About the Difference Between Two Population Means: Matched Samples Critical Value Approach 4. Determine the critical value and rejection rule. For = .05 and df = 9, t.025 = 2.262. Reject H0 if t > 2.262 5. Determine whether to reject H0. Because t = 2.94 > 2.262, we reject H0. We are at least 95% confident that there is a difference in mean delivery times for the two services? © 2005 Thomson/South-Western Slide 41 Inferences About the Difference Between Two Population Proportions Interval Estimation of p1 - p2 Hypothesis Tests About p1 - p2 © 2005 Thomson/South-Western Slide 42 Sampling Distribution of p1 p2 Expected Value E ( p1 p2 ) p1 p2 Standard Deviation (Standard Error) s p1 p2 p1 (1 p1 ) p2 (1 p2 ) n1 n2 where: n1 = size of sample taken from population 1 n2 = size of sample taken from population 2 © 2005 Thomson/South-Western Slide 43 Sampling Distribution of p1 p2 If the sample sizes are large, the sampling distribution of p1 p2 can be approximated by a normal probability distribution. The sample sizes are sufficiently large if all of these conditions are met: n1p1 > 5 n1(1 - p1) > 5 n2p2 > 5 n2(1 - p2) > 5 © 2005 Thomson/South-Western Slide 44 Sampling Distribution of p1 p2 s p1 p2 p1 (1 p1 ) p2 (1 p2 ) n1 n2 p1 p2 p1 – p2 © 2005 Thomson/South-Western Slide 45 Interval Estimation of p1 - p2 Interval Estimate p1 p2 z / 2 © 2005 Thomson/South-Western p1 (1 p1 ) p2 (1 p2 ) n1 n2 Slide 46 Interval Estimation of p1 - p2 Example: Market Research Associates 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. © 2005 Thomson/South-Western Slide 47 Interval Estimation of p1 - p2 Example: Market Research Associates 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? © 2005 Thomson/South-Western Slide 48 Point Estimator of the Difference Between Two Population Proportions 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 p1 = sample proportion of households “aware” of the product after the new campaign p2 = sample proportion of households “aware” of the product before the new campaign 120 60 p1 p2 .48 .40 .08 250 150 © 2005 Thomson/South-Western Slide 49 Interval Estimation of p1 - p2 For = .05, z.025 = 1.96: .48(.52) .40(.60) .48 .40 1.96 250 150 .08 + 1.96(.0510) .08 + .10 Hence, the 95% confidence interval for the difference in before and after awareness of the product is -.02 to +.18. © 2005 Thomson/South-Western Slide 50 Hypothesis Tests about p1 - p2 Hypotheses We focus on tests involving no difference between the two population proportions (i.e. p1 = p2) H0 : p1 p2 0 H a : p1 p2 0 Left-tailed H H00:: H Ha:: a pp1 - pp2 < 00 1 2 pp1 - pp2 > 00 1 2 Right-tailed © 2005 Thomson/South-Western H0 : p1 p2 0 H a : p1 p2 0 Two-tailed Slide 51 Hypothesis Tests about p1 - p2 Pooled Estimate of Standard Error of p1 p2 s p1 p2 1 1 p(1 p) n1 n2 where: n1 p1 n2 p2 p n1 n2 © 2005 Thomson/South-Western Slide 52 Hypothesis Tests about p1 - p2 Test Statistic z ( p1 p2 ) 1 1 p(1 p ) n n 2 1 © 2005 Thomson/South-Western Slide 53 Hypothesis Tests about p1 - p2 Example: Market Research Associates 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? © 2005 Thomson/South-Western Slide 54 Hypothesis Tests about p1 - p2 p -Value and Critical Value Approaches 1. Develop the hypotheses. H0: p1 - p2 < 0 Ha: p1 - p2 > 0 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 © 2005 Thomson/South-Western Slide 55 Hypothesis Tests about p1 - p2 p -Value and Critical Value Approaches 2. Specify the level of significance. = .05 3. Compute the value of the test statistic. 250(. 48) 150(. 40) 180 p . 45 250 150 400 s p1 p2 . 45(. 55)( 1 z 1 ) . 0514 250 150 (.48 .40) 0 .08 1.56 .0514 .0514 © 2005 Thomson/South-Western Slide 56 Hypothesis Tests about p1 - p2 p –Value Approach 4. Compute the p –value. For z = 1.56, the p–value = .0594 5. Determine whether to reject H0. Because p–value > = .05, we cannot reject H0. We cannot conclude that the proportion of households aware of the client’s product increased after the new campaign. © 2005 Thomson/South-Western Slide 57 Hypothesis Tests about p1 - p2 Critical Value Approach 4. Determine the critical value and rejection rule. For = .05, z.05 = 1.645 Reject H0 if z > 1.645 5. Determine whether to reject H0. Because 1.56 < 1.645, we cannot reject H0. We cannot conclude that the proportion of households aware of the client’s product increased after the new campaign. © 2005 Thomson/South-Western Slide 58 End of Chapter 10 © 2005 Thomson/South-Western Slide 59