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1 1 © 2009 Econ-2030(Dr. Tadesse) Chapter 10 Comparisons Involving Means n Inferences About the Difference Between Two Population Means: when  1 and 

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1 1 1 © 2009 Econ-2030(Dr. Tadesse) Chapter 10 Comparisons Involving Means n Inferences About the Difference Between Two Population Means: when  1 and  2 Known Two Population Means: when  1 and  2 Known n Inferences About the Difference Between Two Population Means: Matched Samples Two Population Means: Matched Samples n Inferences About the Difference Between Two Population Means: when  1 and  2 Unknown Two Population Means: when  1 and  2 Unknown n Introduction to Analysis of Variance (Inference about the difference between more than two population means

2 2 2 © 2009 Econ-2030(Dr. Tadesse) Inferences About the Difference Between Two Population Means:  1 and  2 Known Interval Estimation of  1 –  2 Interval Estimation of  1 –  2 Hypothesis Tests About  1 –  2 Hypothesis Tests About  1 –  2

3 3 3 © 2009 Econ-2030(Dr. Tadesse) Estimating the Difference Between Two Population Means Let  1 equal the mean of population 1 and  2 equal Let  1 equal the mean of population 1 and  2 equal the mean of population 2. the mean of population 2. n The difference between the two population means is  1 -  2.  1 -  2. To estimate  1 -  2, we will select a simple random To estimate  1 -  2, we will select a simple random sample of size n 1 from population 1 and a simple sample of size n 1 from population 1 and a simple random sample of size n 2 from population 2. random sample of size n 2 from population 2. n Let equal the mean of sample 1 and equal the mean of sample 2. mean of sample 2. The point estimator of the difference between the The point estimator of the difference between the means of the populations 1 and 2 is. means of the populations 1 and 2 is.

4 4 4 © 2009 Econ-2030(Dr. Tadesse) n Expected Value Sampling Distribution of n Standard Deviation (Standard Error) where:  1 = standard deviation of population 1  2 = standard deviation of population 2  2 = standard deviation of population 2 n 1 = sample size from population 1 n 1 = sample size from population 1 n 2 = sample size from population 2 n 2 = sample size from population 2

5 5 5 © 2009 Econ-2030(Dr. Tadesse) n Interval Estimate Interval Estimation of  1 -  2 :  1 and  2 Known where: 1 -  is the confidence coefficient 1 -  is the confidence coefficient

6 6 6 © 2009 Econ-2030(Dr. Tadesse) n Example: Interval Estimation of  1 -  2 :  1 and  2 Known In a test of driving distance using a mechanical 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. Par, Inc. is a manufacturer Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.”

7 7 7 © 2009 Econ-2030(Dr. Tadesse) Interval Estimation of  1 -  2 :  1 and  2 Known Sample Size Sample Mean Sample #1 Par, Inc. Sample #2 Rap, Ltd. 120 balls 80 balls 120 balls 80 balls 275 yards 258 yards Based on data from previous driving distance Based on data from previous driving distance tests, the two population standard deviations are known with  1 = 15 yards and  2 = 20 yards.

8 8 8 © 2009 Econ-2030(Dr. Tadesse) Interval Estimation of  1 -  2 :  1 and  2 Known Develop a 95% confidence interval estimate of the difference between the mean driving distances of the two brands of golf ball.

9 9 9 © 2009 Econ-2030(Dr. Tadesse) Estimating the Difference Between Two Population Means  1 –  2 = difference between the mean distances the mean distances x 1 - x 2 = Point Estimate of  1 –  2 Population 1 Par, Inc. Golf Balls  1 = mean driving distance of Par distance of Par golf balls Population 1 Par, Inc. Golf Balls  1 = mean driving distance of Par distance of Par golf balls Population 2 Rap, Ltd. Golf Balls  2 = mean driving distance of Rap distance of Rap golf balls Population 2 Rap, Ltd. Golf Balls  2 = mean driving distance of Rap distance of Rap golf balls Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for the Rap golf balls for the Rap golf balls Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for the Rap golf balls for the Rap golf balls Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for the Par golf balls for the Par golf balls Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for the Par golf balls for the Par golf balls

10 10 © 2009 Econ-2030(Dr. Tadesse) Point Estimate of  1 -  2 Point estimate of  1   2 = where:  1 = mean distance for the population of Par, Inc. golf balls of Par, Inc. golf balls  2 = mean distance for the population of Rap, Ltd. golf balls of Rap, Ltd. golf balls = 275  258 = 17 yards

11 11 © 2009 Econ-2030(Dr. Tadesse) Interval Estimation of  1 -  2 :   1 and   2 Known We are 95% confident that the difference between 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. 17 + 5.14 or 11.86 yards to 22.14 yards

12 12 © 2009 Econ-2030(Dr. Tadesse) Hypothesis Tests About  1   2 :  1 and  2 Known Hypothesis Testing Hypothesis Testing Left-tailed Right-tailedTwo-tailed Test Statistic Test Statistic

13 13 © 2009 Econ-2030(Dr. Tadesse) Hypothesis Tests About  1   2 :  1 and  2 Known Can we conclude, using 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?

14 14 © 2009 Econ-2030(Dr. Tadesse) H 0 :  1 -  2 < 0  H a :  1 -  2 > 0 where:  1 = mean distance for the population of Par, Inc. golf balls of Par, Inc. golf balls  2 = mean distance for the population of Rap, Ltd. golf balls of Rap, Ltd. golf balls 1. Develop the hypotheses. Hypothesis Tests About  1   2 :  1 and  2 Known 2. Specify the level of significance.  =.01

15 15 © 2009 Econ-2030(Dr. Tadesse) 3. Compute the value of the test statistic. Hypothesis Tests About  1   2 :  1 and  2 Known

16 16 © 2009 Econ-2030(Dr. Tadesse) Hypothesis Tests About  1   2 :  1 and  2 Known 5. Compare the Test Statistic with the Critical Value. Because z = 6.49 > 2.33, we reject H 0. Using the Critical Value Approach Using the Critical Value Approach For  =.01, z.01 = 2.33 4. Determine the critical value and rejection rule. The sample evidence indicates the mean driving 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 17 © 2009 Econ-2030(Dr. Tadesse) Using the p –Value Approach Using the p –Value Approach 4. Compute the p –value. For z = 6.49, the p –value <.0001. Hypothesis Tests About  1   2 :  1 and  2 Known 5. Determine whether to reject H 0. Because p –value <  =.01, we reject H 0. At the.01 level of significance, the sample evidence 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.

18 18 © 2009 Econ-2030(Dr. Tadesse) Inferences About the Difference Between Two Population Means:  1 and  2 Unknown Interval Estimation of  1 –  2 Interval Estimation of  1 –  2 Hypothesis Tests About  1 –  2 Hypothesis Tests About  1 –  2

19 19 © 2009 Econ-2030(Dr. Tadesse) Interval Estimation of  1 -  2 :  1 and  2 Unknown When  1 and  2 are unknown, we will: replace z  /2 with t  /2. replace z  /2 with t  /2. we use the sample standard deviations s 1 and s 2 we use the sample standard deviations s 1 and s 2 as estimates of  1 and  2, and

20 20 © 2009 Econ-2030(Dr. Tadesse) Where the degrees of freedom for t  /2 are: Interval Estimation of  1 -  2 :  1 and  2 Unknown n Interval Estimate

21 21 © 2009 Econ-2030(Dr. Tadesse) n Example Difference Between Two Population Means:  1 and  2 Unknown Specific Motors of Detroit 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.

22 22 © 2009 Econ-2030(Dr. Tadesse) Difference Between Two Population Means:  1 and  2 Unknown Sample Size Sample Mean Sample Std. Dev. Sample #1 M Cars Sample #2 J Cars 24 cars 2 8 cars 24 cars 2 8 cars 29.8 mpg 27.3 mpg 2.56 mpg 1.81 mpg

23 23 © 2009 Econ-2030(Dr. Tadesse) Difference Between Two Population Means:  1 and  2 Unknown Develop a 90% confidence interval estimate of the difference between the mpg performances of the two models of automobile.

24 24 © 2009 Econ-2030(Dr. Tadesse) Point estimate of  1   2 = Point Estimate of  1   2 where:  1 = mean miles-per-gallon for the population of M cars population of M cars  2 = mean miles-per-gallon for the population of J cars population of J cars = 29.8 - 27.3 = 2.5 mpg

25 25 © 2009 Econ-2030(Dr. Tadesse) Interval Estimation of  1   2 :  1 and  2 Unknown The degrees of freedom for t  /2 are: With  /2 =.05 and df = 24, t  /2 = 1.711

26 26 © 2009 Econ-2030(Dr. Tadesse) Interval Estimation of  1   2 :  1 and  2 Unknown We are 90% confident that the difference between 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. 2.5 + 1.069 or 1.431 to 3.569 mpg

27 27 © 2009 Econ-2030(Dr. Tadesse) Hypothesis Tests About  1   2 :  1 and  2 Unknown n Hypothesis Testing Left-tailedRight-tailedTwo-tailed n Test Statistic

28 28 © 2009 Econ-2030(Dr. Tadesse) Hypothesis Tests About  1   2 :  1 and  2 Unknown Can we conclude, using a.05 level of significance, that the 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?

29 29 © 2009 Econ-2030(Dr. Tadesse) H 0 :  1 -  2 = 0  H a :  1 -  2 > 0 where:  1 = mean mpg for the population of M cars  2 = mean mpg for the population of J cars 1. Develop the hypotheses. Hypothesis Tests About  1   2 :  1 and  2 Unknown

30 30 © 2009 Econ-2030(Dr. Tadesse) 2. Specify the level of significance. 3. Compute the value of the test statistic.  =.05 Hypothesis Tests About  1   2 :  1 and  2 Unknown

31 31 © 2009 Econ-2030(Dr. Tadesse) Hypothesis Tests About  1   2 :  1 and  2 Unknown Using the p –Value Approach Using the p –Value Approach 4. Compute the p –value. Compute the the degrees of freedom:

32 32 © 2009 Econ-2030(Dr. Tadesse) 4. Determine the critical value and rejection rule. Using the Critical Value Approach Using the Critical Value Approach Hypothesis Tests About  1   2 :  1 and  2 Unknown For  =.05 and df = 24, t.05 = 1.711

33 33 © 2009 Econ-2030(Dr. Tadesse) Using the Critical Value Approach Using the Critical Value Approach Hypothesis Tests About  1   2 :  1 and  2 Unknown 5. Compare the Test Statistic with the Critical Value. As t he test statistic 4.003 is greater than the Critical value 1.711, we reject H 0. 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?. 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?.

34 34 © 2009 Econ-2030(Dr. Tadesse) 5. Compute the P-value and determine whether to reject H 0. 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?. 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?. As the p –value <  =.05, we reject H 0. Hypothesis Tests About  1   2 :  1 and  2 Unknown Using the p-Value Approach Using the p-Value Approach

35 35 © 2009 Econ-2030(Dr. Tadesse) In a matched-sample design each sampled item In a matched-sample design each sampled item provides a pair of data values. provides a pair of data values. This design often leads to a smaller sampling error This design often leads to a smaller sampling error than the independent-sample design because than the independent-sample design because variation between sampled items is eliminated as a variation between sampled items is eliminated as a source of sampling error. source of sampling error. Inferences About the Difference Between Two Population Means: Matched Samples

36 36 © 2009 Econ-2030(Dr. Tadesse) n Example: A Chicago-based firm has 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. Inferences About the Difference Between Two Population Means: Matched Samples

37 37 © 2009 Econ-2030(Dr. Tadesse) In testing the delivery times 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. Inferences About the Difference Between Two Population Means: Matched Samples

38 38 © 2009 Econ-2030(Dr. Tadesse) 3230191615181410 71625241515131515 8 911 UPXINTEXDifference District Office Seattle Los Angeles BostonCleveland New York HoustonAtlanta St. Louis MilwaukeeDenver Delivery Time (Hours) 7 6 4 1 2 3 2-25 Inferences About the Difference Between Two Population Means: Matched Samples

39 39 © 2009 Econ-2030(Dr. Tadesse) H 0 :  d = 0  H a :  d  Let  d = the mean of the difference values for the two delivery services for the population two delivery services for the population of district offices of district offices 1. Develop the hypotheses. Inferences About the Difference Between Two Population Means: Matched Samples

40 40 © 2009 Econ-2030(Dr. Tadesse) 2. Specify the level of significance.  =.05 3. Compute the value of the test statistic. Inferences About the Difference Between Two Population Means: Matched Samples

41 41 © 2009 Econ-2030(Dr. Tadesse) 4. Determine the critical value and rejection rule. Using the Critical Value Approach Using the Critical Value Approach For  =.05 and df = 9, t.025 = 2.262. 5. Compare the Test Statistic with the Critical Value Because t = 2.94 > 2.262, we reject H 0. We are at least 95% confident that there is a difference in mean delivery times for the two services? Inferences About the Difference Between Two Population Means: Matched Samples

42 42 © 2009 Econ-2030(Dr. Tadesse) 5. Determine whether to reject H 0. We are at least 95% confident that there is a difference in mean delivery times for the two services? We are at least 95% confident that there is a difference in mean delivery times for the two services? 4. Compute the p –value. For t = 2.94 and df = 9, the p –value is between 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.) Because p –value <  =.05, we reject H 0. Using the p –Value Approach Using the p –Value Approach Inferences About the Difference Between Two Population Means: Matched Samples

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