. SLIDES. BY John Loucks St. Edward s University 1
Chapter 10, Part A Inference About Means and Proportions with Two Populations n Inferences About the Difference Between Two Population Means: σ 1 and σ Known n Inferences About the Difference Between Two Population Means: σ 1 and σ Unknown n Inferences About the Difference Between Two Population Means: Matched Samples
Inferences About the Difference Between Two Population Means: σ 1 and σ Known n Interval Estimation of µ 1 µ n Hypothesis Tests About µ 1 µ 3
Estimating the Difference Between Two Population Means n Let µ 1 equal the mean of population 1 and µ equal the mean of population. n The difference between the two population means is µ 1 - µ. n To estimate µ 1 - µ, we will select a simple random sample of size n 1 from population 1 and a simple random sample of size n from population. n Let x 1 equal the mean of sample 1 and x equal the mean of sample. The point estimator of the difference between the means of the populations 1 and is. x x 1 4
Sampling Distribution of x n Expected Value E( x x ) = µ µ 1 1 x 1 n Standard Deviation (Standard Error) σ x σ1 σ x = + n n 1 where: σ 1 = standard deviation of population 1 σ = standard deviation of population n 1 = sample size from population 1 n = sample size from population 1 5
n Interval Estimate Interval Estimation of µ 1 - µ : σ 1 and σ Known σ σ x1 x ± zα / + n n 1 1 where: 1 - α is the confidence coefficient 6
Interval Estimation of µ 1 - µ : σ 1 and σ Known n 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 µ 1 - µ : σ 1 and σ Known n Example: Par, Inc. Sample Size Sample Mean Sample #1 Par, Inc. Sample # Rap, Ltd. 10 balls 80 balls 75 yards 58 yards Based on data from previous driving distance tests, the two population standard deviations are known with σ 1 = 15 yards and σ = 0 yards. 8
Interval Estimation of µ 1 - µ : σ 1 and σ Known n 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
Estimating the Difference Between Two Population Means Population 1 Par, Inc. Golf Balls µ 1 = mean driving distance of Par golf balls m 1 µ = difference between the mean distances Population Rap, Ltd. Golf Balls µ = mean driving distance of Rap golf balls Simple random sample of n 1 Par golf balls x 1 = sample mean distance for the Par golf balls Simple random sample of n Rap golf balls x = sample mean distance for the Rap golf balls x 1 - x = Point Estimate of m 1 µ 10
Point Estimate of µ 1 - µ Point estimate of µ 1 µ = x where: x 1 = 75 58 = 17 yards µ 1 = mean distance for the population of Par, Inc. golf balls µ = mean distance for the population of Rap, Ltd. golf balls 11
Interval Estimation of µ 1 - µ : σ 1 and σ Known σ1 σ ( x1 x z 17 1 96 15 ) ( 0) ± α/ + = ±. + n n 10 80 1 17 + 5.14 or 11.86 yards to.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.14 yards. 1
Hypotheses Hypothesis Tests About µ 1 µ : σ 1 and σ Known H H : µ µ D 0 1 0 : a µ 1 µ < D0 H H : µ µ D 0 1 0 : a µ 1 µ > D0 H H : µ µ = D µ µ D 0 1 0 : a 1 0 Left-tailed Right-tailed Two-tailed Test Statistic z = ( x x ) D 1 0 σ n σ + n 1 1 13
Hypothesis Tests About µ 1 µ : σ 1 and σ Known n Example: Par, Inc. Can we conclude, using α =.01, that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls? 14
Hypothesis Tests About µ 1 µ : σ 1 and σ Known p Value and Critical Value Approaches 1. Develop the hypotheses. H 0 : µ 1 - µ < 0 H a : µ 1 - µ > 0 where: µ 1 = mean distance for the population of Par, Inc. golf balls µ = mean distance for the population of Rap, Ltd. golf balls. Specify the level of significance. α =.01 15
Hypothesis Tests About µ 1 µ : σ 1 and σ Known p Value and Critical Value Approaches 3. Compute the value of the test statistic. z z = ( x x ) D 1 0 σ n σ + n 1 1 (35 18) 0 17 = = = (15) (0).6 + 10 80 6.49 16
Hypothesis Tests About µ 1 µ : σ 1 and σ Known p Value Approach 4. Compute the p value. For z = 6.49, the p value <.0001. 5. Determine whether to reject H 0. Because p value < α =.01, we reject H 0. 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. 17
Hypothesis Tests About µ 1 µ : σ 1 and σ Known Critical Value Approach 4. Determine the critical value and rejection rule. For α =.01, z.01 =.33 Reject H 0 if z >.33 5. Determine whether to reject H 0. Because z = 6.49 >.33, we reject H 0. 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. 18
Inferences About the Difference Between Two Population Means: σ 1 and σ Unknown n Interval Estimation of µ 1 µ n Hypothesis Tests About µ 1 µ 19
Interval Estimation of µ 1 - µ : σ 1 and σ Unknown When σ 1 and σ are unknown, we will: use the sample standard deviations s 1 and s as estimates of σ 1 and σ, and replace z α/ with t α/. 0
n Interval Estimate Interval Estimation of µ 1 - µ : σ 1 and σ Unknown s x1 x ± tα / + n s n 1 1 Where the degrees of freedom for t α/ are: df = s n s + n 1 1 1 s 1 s 1 + n1 1 n1 n 1 n 1
n Difference Between Two Population Means: σ 1 and σ Unknown Example: Specific Motors Specific Motors of Detroit has developed a new Automobile known as the M car. 4 M cars and 8 J cars (from Japan) were road tested to compare milesper-gallon (mpg) performance. The sample statistics are shown on the next slide.
Difference Between Two Population Means: σ 1 and σ Unknown n Example: Specific Motors Sample #1 M Cars Sample # J Cars 4 cars 8 cars 9.8 mpg 7.3 mpg.56 mpg 1.81 mpg Sample Size Sample Mean Sample Std. Dev. 3
Difference Between Two Population Means: σ 1 and σ Unknown n Example: Specific Motors Let us develop a 90% confidence interval estimate of the difference between the mpg performances of the two models of automobile. 4
Point estimate of µ 1 µ = x where: Point Estimate of µ 1 µ x 1 = 9.8-7.3 =.5 mpg µ 1 = mean miles-per-gallon for the population of M cars µ = mean miles-per-gallon for the population of J cars 5
Interval Estimation of µ 1 µ : σ 1 and σ Unknown The degrees of freedom for t α/ are: df (.56) (1.81) + 4 8 = = 4.07 = 4 1 (.56) 1 (1.81) + 4 1 4 8 1 8 With α/ =.05 and df = 4, t α/ = 1.711 6
Interval Estimation of µ 1 µ : σ 1 and σ Unknown s s (.56) (1.81) x1 x ± tα / + = 9.8 7.3 ± 1.711 + n n 4 8 1 1.5 + 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. 7
n Hypotheses H : Ha: Hypothesis Tests About µ 1 µ : σ 1 and σ Unknown µ µ D µ µ < D 0 1 0 1 0 H : Ha: µ µ D µ µ > D 0 1 0 1 0 H H : µ µ = D µ µ D 0 1 0 : a 1 0 Left-tailed Right-tailed Two-tailed n Test Statistic t = ( x x ) D 1 0 s n s n 1 + 1 8
Hypothesis Tests About µ 1 µ : σ 1 and σ Unknown n 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? 9
Hypothesis Tests About µ 1 µ : σ 1 and σ Unknown p Value and Critical Value Approaches 1. Develop the hypotheses. H 0 : µ 1 - µ < 0 H a : µ 1 - µ > 0 where: µ 1 = mean mpg for the population of M cars µ = mean mpg for the population of J cars 30
Hypothesis Tests About µ 1 µ : σ 1 and σ Unknown p Value and Critical Value Approaches. Specify the level of significance. α =.05 3. Compute the value of the test statistic. t ( x x ) D (9.8 7.3) 0 1 0 = = = s1 s (.56) (1.81) n + + n 4 8 1 4.003 31
Hypothesis Tests About µ 1 µ : σ 1 and σ Unknown p Value Approach 4. Compute the p value. The degrees of freedom for t α are: df (.56) (1.81) + 4 8 = = 40.566 = 41 1 (.56) 1 (1.81) + 4 1 4 8 1 8 Because t = 4.003 > t.005 = 1.683, the p value <.005. 3
Hypothesis Tests About µ 1 µ : σ 1 and σ Unknown p Value Approach 5. Determine whether to reject H 0. Because p value < α =.05, we reject H 0. 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?. 33
Hypothesis Tests About µ 1 µ : σ 1 and σ Unknown Critical Value Approach 4. Determine the critical value and rejection rule. For α =.05 and df = 41, t.05 = 1.683 Reject H 0 if t > 1.683 5. Determine whether to reject H 0. Because 4.003 > 1.683, we reject H 0. 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?. 34
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. 35
Inferences About the Difference Between Two Population Means: Matched Samples n 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. 36
Inferences About the Difference Between Two Population Means: Matched Samples n 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. 37
Inferences About the Difference Between Two Population Means: Matched Samples District Office Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee Denver Delivery Time (Hours) UPX INTEX Difference 3 30 19 16 15 18 14 10 7 16 5 4 15 15 13 15 15 8 9 11 7 6 4 1 3-1 - 5 38
Inferences About the Difference Between Two Population Means: Matched Samples p Value and Critical Value Approaches 1. Develop the hypotheses. H 0 : µ d = 0 H a : µ d 0 Let µ d = the mean of the difference values for the two delivery services for the population of district offices 39
Inferences About the Difference Between Two Population Means: Matched Samples p Value and Critical Value Approaches. Specify the level of significance. α =.05 3. Compute the value of the test statistic. d di = = n ( 7 + 6+... + 5) = 10. 7 s d = ( di d ) 76. 1 = = n 1 9 d µ d.7 0 t = = = s n.9 10 d. 9.94 40
Inferences About the Difference Between Two Population Means: Matched Samples p Value Approach 4. Compute the p value. For t =.94 and df = 9, the p value is between.0 and.01. (This is a two-tailed test, so we double the upper-tail areas of.01 and.005.) 5. Determine whether to reject H 0. Because p value < α =.05, we reject H 0. We are at least 95% confident that there is a difference in mean delivery times for the two services? 41
Inferences About the Difference Between Two Population Means: Matched Samples Critical Value Approach 4. Determine the critical value and rejection rule. For α =.05 and df = 9, t.05 =.6. Reject H 0 if t >.6 5. Determine whether to reject H 0. Because t =.94 >.6, we reject H 0. We are at least 95% confident that there is a difference in mean delivery times for the two services? 4
End of Chapter 10 Part A 43