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Copy of MGMT305-Lec7-Fall03 - Chapter 10 Comparisons...

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Unformatted text preview: Chapter 10 Comparisons Involving Means Estimation of the Difference Between the Means of Two Populations: Independent Samples s Hypothesis Tests about the Difference between the Means of Two Populations: Independent Samples s Inferences about the Difference between the Means of Two Populations: Matched Samples s = µ1 ? µ2 Slide 1 Estimation of the Difference Between the Means of Two Populations: Independent Samples s s s s Point Estimator of the Difference between the Means of Two Populations x1 − x2 Sampling Distribution of Interval Estimate of µ1 −µ2 : Large­Sample Case Large­Sample Case Interval Estimate of µ1 −µ2 : Small­Sample Case Small­Sample Case Slide 2 Point Estimator of the Difference Between the Means of Two Populations s s s s s Let µ1 equal the mean of population 1 and µ2 equal the mean of population 2. The difference between the two population means is µ1 ­ µ 2. To estimate µ1 ­ µ2, 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 x1 Let equal the mean of sample 1 and equal the mean of sample 2. The point estimator of the difference between the means x1 − x2 of the populations 1 and 2 is . Slide 3 Sampling Distribution of x1 − x2 s Properties of the Sampling Distribution of • Expected Value x1 − x2 E ( x1 − x2 ) = µ1 − µ 2 • Standard Deviation σ x1 − x2 2 σ1 σ 2 = +2 n1 n2 where: σ1 = standard deviation of population 1 σ2 = standard deviation of population 2 n1 = sample size from population 1 n2 = sample size from population 2 Slide 4 100(1 ­ α) % Confidence Interval of µ 1 ­ µ 2: Large­Sample Case (n1 > 30 and n2 > 30) s Interval Estimate with σ1 and σ2 Known where: s x1 − x2 ± zα / 2 σ x1 − x2 1 ­ α is the confidence coefficient Interval Estimate with σ1 and σ2 Unknown x1 − x2 ± zα / 2 sx1 − x2 where: sx1 − x2 2 2 s1 s2 = + n1 n2 Slide 5 Example: Par, Inc. s Interval Estimate of µ1 ­ µ2: Large­Sample 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 6 Example: Par, Inc. s Interval Estimate of µ1 ­ µ2: Large­Sample Case • Sample Statistics Sample #1 Sample #2 Par, Inc. Rap, Ltd. Sample Size n1 = 120 balls n2 = 80 balls Mean Standard Dev. x x1 2 = 235 yards = 218 yards s1 = 15 yards s2 = 20 yards Slide 7 Example: Par, Inc. s Point Estimate of the Difference Between Two Population Means µ1 = mean distance for the population of Par, Inc. golf balls µ2 = mean distance for the population of Rap, Ltd. golf balls (a) What is the point estimate of µ1 ­ µ2? Slide 8 Point Estimator of the Difference Between the Means of Two Populations Population 1 Population 2 Population 1 Population 2 Par, Inc. Golf Balls Rap, Ltd. Golf Balls Par, Inc. Golf Balls µ1 = mean driving µ2 = mean driving 1 = mean driving 2 distance of Par distance of Rap golf balls golf balls golf balls µ – µ2 = difference between = difference between 1 the mean distances Simple random sample of n1 Par golf balls of n1 Par golf balls Simple random sample of n2 Rap golf balls n2 x1 = sample mean distance x1 = sample mean distance for sample of Par golf ball for sample of Par golf ball x2 = sample mean distance x2 = sample mean distance for sample of Rap golf ball for sample of Rap golf ball x1 ­ x2 = Point Estimate of µ – µ2 1 Slide 9 Example: Par, Inc. s (b) Compute the 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large­Sample Case, σ1 and σ2 Unknown 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 yards. Slide 10 100(1 ­ α) % Confidence Interval of µ 1 ­ µ 2: Small­Sample Case (n1 < 30 and/or n2 < 30) s Assume both populations have normal distributions. s Interval Estimate with σ1 and σ2 Known x1 − x2 ± zα / 2 σ x1 − x2 where: σ x1 − x2 2 σ1 σ 2 = +2 n1 n2 Slide 11 100(1 ­ α) % Confidence Interval of µ 1 ­ µ 2: Small­Sample Case (n1 < 30 and/or n2 < 30) s If σ1 and σ2 Unknown, make following assumptions: • • s s Both populations have normal distributions 2 2 σ = σ2 = σ 2 Variances are equal: 1 11 Then σ x1 − x2 = σ 2 ( + ) n1 n2 Interval Estimate with σ1 and σ2 Unknown x1 − x2 ± tα / 2 sx1 − x2 where: sx1 − x2 11 = s( + ) n1 n2 2 2 2 ( n1 − 1) s1 + ( n2 − 1) s2 s2 = n1 + n2 − 2 Slide 12 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 miles­per­ gallon (mpg) performance. The sample statistics are: Sample Size Mean Standard Deviation Sample #1 Sample #2 M Cars J Cars n1 = 12 cars n2 = 8 cars x1 x2 = 29.8 mpg = 27.3 mpg s1 = 2.56 mpg s2 = 1.81 mpg Slide 13 Example: Specific Motors s Point Estimate of the Difference Between Two Population Means µ1 = mean miles­per­gallon for the population of M cars µ2 = mean miles­per­gallon for the population of J cars (a) What is the point estimate of µ1 ­ µ2? Slide 14 Example: Specific Motors s 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small­Sample 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 σ 2. Slide 15 Example: Specific Motors s (b) Construct the 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small­ Sample Case We are 95% confident that the difference between the mean mpg ratings of the two car types is from mpg . Slide 16 Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples (Large­Sample Case) s Hypotheses H0: µ1 ­ µ2 < 0 H0: µ1 ­ µ2 > 0 H0: µ1 ­ µ2 = 0 Ha: µ1 ­ µ2 > 0 Ha: µ1 ­ µ2 < 0 Ha: µ1 ­ µ2 ≠ 0 Test Statistic Large­Sample Case (n1 > 30 and n2 > 30) s σ1, σ2 known σ1, σ2 unknown ( x1 − x2 ) − ( µ1 − µ 2 ) ( x1 − x2 ) − ( µ1 − µ 2 ) z= z= 2 2 σ1 n1 + σ 2 n2 s12 n1 + s2 n2 2 Slide 17 Large­Sample Case (n1 > 30 and n2 > 30) s Rejection Rule Slide 18 Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples (Small­Sample Case) s Test Statistic: Small­Sample Case (n1 < 30 and/or n2 < 30) σ1, σ2 known σ1, σ2 unknown ( x1 − x2 ) − ( µ1 − µ 2 ) ( x1 − x2 ) − ( µ1 − µ 2 ) z= t= 2 2 σ1 n1 + σ 2 n2 s2 (1 n1 + 1 n2 ) where s 2 2 ( n1 − 1) s1 + ( n2 − 1) s2 s2 = n1 + n2 − 2 If σ1 and σ2 Unknown, make the assumptions: • • Both populations have normal distributions σ 12 = σ 22 = σ 2 Variances are equal: Slide 19 Small­Sample Case (n1 < 30 and/or n2 < 30) s Rejection Rule Slide 20 Example: Par, Inc. s Hypothesis Tests About the Difference Between the Means of Two Populations: Large­Sample 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 21 Example: Par, Inc. s Hypothesis Tests About the Difference Between the Means of Two Populations: Large­Sample Case • Sample Statistics Sample #1 Par, Inc. Sample Size n1 = 120 balls Mean Standard Dev. Sample #2 Rap, Ltd. n2 = 80 balls x1 = 235 yards s1 = 15 yards x2 = 218 yards s2 = 20 yards Slide 22 Example: Par, Inc. Hypothesis Tests About the Difference Between the Means of Two Populations: Large­Sample Case Q. 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? µ1 = mean distance for the population of Par, Inc. s golf balls µ2 = mean distance for the population of Rap, Ltd. golf balls (a) What are the Hypotheses H0 and Ha? Slide 23 Example: Par, Inc. s Hypothesis Tests About the Difference Between the Means of Two Populations: Large­Sample Case (b) Compute the Test Statistic (c) What is the conclusion? Slide 24 Example: Specific Motors Specific Motors of Detroit has developed a new automobile known as the M car. 12 M cars and 8 J cars (fromJapan) were road tested to compare miles­per­ gallon (mpg) performance. The sample statistics are: Sample Size Mean Standard Deviation Sample #1 Sample #2 M Cars J Cars n1 = 12 cars n2 = 8 cars x1 x2 = 29.8 mpg = 27.3 mpg s1 = 2.56 mpg s2 = 1.81 mpg Slide 25 Example: Specific Motors Hypothesis Tests About the Difference Between the Means of Two Populations: Small­Sample Case Q. Can we conclude, using a .05 level of significance, that the miles­per­gallon (mpg) performance of M cars is greater than the miles­per­gallon performance of J cars? µ1 = mean mpg for the population of M cars s µ2 = mean mpg for the population of J cars (a) What are the Hypotheses H0 and Ha? Slide 26 Example: Specific Motors s Hypothesis Tests About the Difference Between the Means of Two Populations: Small­Sample Case (b) Find the Test Statistic (c) What is the conclusion? Slide 27 Inference About the Difference Between the Means of Two Populations: Matched Samples s s s With a matched­sample design each sampled item provides a pair of data values. The matched­sample design can be referred to as blocking. This design often leads to a smaller sampling error than the independent­sample design because variation between sampled items is eliminated as a source of sampling error. Slide 28 Example: Express Deliveries s Inference About the Difference Between the Means of Two Populations: Matched Samples A Chicago­based 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 29 Example: Express Deliveries District Office Delivery Time (Hours) UPX INTEX Difference Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee 7 Denver 32 25 30 24 19 15 16 15 15 13 18 15 14 15 10 8 9 16 11 Slide 30 Example: Express Deliveries s Inference About the Difference Between the Means of Two Populations: Matched Samples Let µd = the mean of the difference values for the two delivery services for the population of district offices • • Hypotheses H0: µd = 0, Ha: µd ≠ 0 Rejection Rule Assuming the population of difference values is approximately normally distributed, the t distribution with n ­ 1 degrees of freedom applies. With α = .05, t.025 = 2.262 (9 degrees of freedom). Reject H0 if t < ­2.262 or if t > 2.262 Slide 31 Example: Express Deliveries s Inference About the Difference Between the Means of Two Populations: Matched Samples ∑di d= n sd = t= • (di − d ) 2 ∑ n −1 d − µd sd n What is the conclusion? Slide 32 ...
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This note was uploaded on 03/03/2012 for the course MGMT 305 taught by Professor Priya during the Spring '08 term at Purdue University-West Lafayette.

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