Slides by. John Loucks. St. Edward s University. Slide South-Western, a part of Cengage Learning

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1 Slides by John Loucks St. Edward s University Slide 1

2 Chapter 10 Comparisons Involving Means Part A 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 Slide 2

3 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 Slide 3

4 Estimating the Difference Between Two Population Means Let m 1 equal the mean of population 1 and m 2 equal the mean of population 2. The difference between the two population means is m 1 - m 2. To estimate m 1 - m 2, we will select a simple random sample of size n 1 from population 1 and a simple random sample of size n 2 from population 2. x 1 x 2 Let equal the mean of sample 1 and equal the mean of sample 2. The point estimator of the difference between the means of the populations 1 and 2 is. x x 1 2 Slide 4

5 Sampling Distribution of x Expected Value E( x x ) m m Standard Deviation (Standard Error) s x s1 s2 x n n x 1 2 where: s 1 = standard deviation of population 1 s 2 = standard deviation of population 2 n 1 = sample size from population 1 n 2 = sample size from population Slide 5

6 Interval Estimate Interval Estimation of m 1 - m 2 : s 1 and s 2 Known s s x1 x2 z / 2 n n where: 1 - is the confidence coefficient Slide 6

7 Example: Par, Inc. Interval Estimation of m 1 - m 2 : s 1 and s 2 Known 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

8 Example: Par, Inc. Interval Estimation of m 1 - m 2 : s 1 and s 2 Known Sample Size Sample Mean Sample #1 Par, Inc. Sample #2 Rap, Ltd. 120 balls 80 balls 295 yards 278 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. Slide 8

9 Example: Par, Inc. Interval Estimation of m 1 - m 2 : s 1 and s 2 Known Let us develop a 95% confidence interval estimate of the difference between the mean driving distances of the two brands of golf ball. Slide 9

10 Estimating the Difference Between Two Population Means Population 1 Par, Inc. Golf Balls m 1 = mean driving distance of Par golf balls m 1 m 2 = difference between the mean distances Population 2 Rap, Ltd. Golf Balls m 2 = 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 2 Rap golf balls x 2 = sample mean distance for the Rap golf balls x 1 - x 2 = Point Estimate of m 1 m 2 Slide 10

11 Point Estimate of m 1 - m 2 Point estimate of m 1 m 2 = x where: x 1 2 = = 17 yards m 1 = mean distance for the population of Par, Inc. golf balls m 2 = mean distance for the population of Rap, Ltd. golf balls Slide 11

12 Interval Estimation of m 1 - m 2 : s 1 and s 2 Known 2 s1 s2 ( x1 x2 z ) ( 20) /. n n or yards to yards We are 95% confident that the difference between the mean driving distances of Par, Inc. balls and Rap, Ltd. balls is to yards. Slide 12

13 Interval Estimation of m 1 - m 2 : s 1 and s 2 Known Excel Formula Worksheet A B C D E 1 Par Rap Par, Inc. Rap, Ltd Sample Size =COUNT(A2:A121) =COUNT(B2:B81) Sample Mean =AVERAGE(A2:A121) =AVERAGE(B2:B81) Popul. Std. Dev Standard Error =SQRT(D5^2/D2+E5^2/E2) Confid. Coeff Level of Signif. =1-D z Value =NORMSINV(1-D9/2) Margin of Error =D10*D Pt. Est. of Diff. =D3-E Lower Limit =D13-D Upper Limit =D13+D11 Note: Rows are not shown. Slide 13

14 Interval Estimation of m 1 - m 2 : s 1 and s 2 Known Excel Formula Worksheet A B C D E 1 Par Rap Par, Inc. Rap, Ltd Sample Size Sample Mean Popul. Std. Dev Standard Error Confid. Coeff Level of Signif z Value Margin of Error Pt. Est. of Diff Lower Limit Upper Limit Note: Rows are not shown. Slide 14

15 Hypotheses H : Ha: Hypothesis Tests About m 1 m 2: s 1 and s 2 Known m m D m m D H : m m D Ha: m m D H : Ha: m m D m m D Left-tailed Right-tailed Two-tailed Test Statistic z ( x x ) D s n s n Slide 15

16 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? Slide 16

17 Hypothesis Tests About m 1 m 2: s 1 and s 2 Known p Value and Critical Value Approaches 1. Develop the hypotheses. H 0 : m 1 - m 2 < 0 H a : m 1 - m 2 > 0 where: m 1 = mean distance for the population of Par, Inc. golf balls m 2 = mean distance for the population of Rap, Ltd. golf balls 2. Specify the level of significance. =.01 Slide 17

18 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 ( x x ) D s n s n ( ) 0 17 z 2 2 (15) (20) Slide 18

19 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 < Determine whether to reject H 0. Because p value < =.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. Slide 19

20 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 H 0 if z > Determine whether to reject H 0. Because z = 6.49 > 2.33, 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. Slide 20

21 Hypothesis Tests About m 1 m 2: s 1 and s 2 Known Excel s z-test: Two Sample for Means Tool Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose z-test: Two Sample for Means from the list of Analysis Tools continued Slide 21

22 Hypothesis Tests About m 1 m 2: s 1 and s 2 Known Excel Dialog Box Slide 22

23 Hypothesis Tests About m 1 m 2: s 1 and s 2 Known Excel Value Worksheet A B C D E F 1 Par Rap z-test: Two Sample for Means Par, Inc. Rap, Ltd Mean Known Variance Observations Hypothesized Mean Difference z P(Z<=z) one-tail E z Critical one-tail P(Z<=z) two-tail E z Critical two-tail Note: Rows are not shown. Slide 23

24 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 Slide 24

25 Interval Estimation of m 1 - m 2 : s 1 and s 2 Unknown When s 1 and s 2 are unknown, we will: use the sample standard deviations s 1 and s 2 as estimates of s 1 and s 2, and replace z /2 with t /2. Slide 25

26 Interval Estimation of m 1 - m 2 : s 1 and s 2 Unknown Interval Estimate s x1 x2 t / 2 n s n Where the degrees of freedom for t /2 are: df s n s n s2 1 s 1 n1 1 n1 n2 1 n2 2 Slide 26

27 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-pergallon (mpg) performance. The sample statistics are shown on the next slide. Let us develop a 90% confidence interval estimate of the difference between the mpg performances of the two models of automobile. Slide 27

28 Difference Between Two Population Means: s 1 and s 2 Unknown Example: Specific Motors Sample #1 M Cars Sample #2 J Cars 24 cars 28 cars 29.8 mpg 27.3 mpg 2.56 mpg 1.81 mpg Sample Size Sample Mean Sample Std. Dev. Slide 28

29 Point estimate of m 1 m 2 = x where: Point Estimate of m 1 m 2 x 1 2 = = 2.5 mpg m 1 = mean miles-per-gallon for the population of M cars m 2 = mean miles-per-gallon for the population of J cars Slide 29

30 Interval Estimation of m 1 m 2: s 1 and s 2 Unknown The degrees of freedom for t /2 are: df 2 2 (2.56) (1.81) (2.56) 1 (1.81) With /2 =.05 and df = 24, t /2 = Slide 30

31 Interval Estimation of m 1 m 2: s 1 and s 2 Unknown s s (2.56) (1.81) x1 x2 t / n n or to mpg We are 90% confident that the difference between the miles-per-gallon performances of M cars and J cars is to mpg. Slide 31

32 Interval Estimation of m 1 m 2: s 1 and s 2 Unknown Excel Formula Worksheet A B C D E 1 M J M Cars J Cars Sample Size =COUNT(A2:A25) =COUNT(B2:B29) Sample Mean =AVERAGE(A2:A25) =AVERAGE(B2:B29) Sample Std. Dev. =STDEV(A2:A25) =STDEV(B2:B29) Est. of Variance =D4^2/D2+E4^2/E Standard Error =SQRT(D6) Confid. Coeff Level of Signif. =1-D Degr. of Freedom =D6^2/((1/(D2-1))*(D4^2/D2)^2+(1/(E2-1))*(E4^2/E2)^2)) t Value =TINV(D10,D11) Margin of Error =D12*D Point Est. of Diff. =D3-E Lower Limit =D15-D Upper Limit =D15+D13 Note: Rows are not shown. Slide 32

33 Interval Estimation of m 1 m 2: s 1 and s 2 Unknown Excel Formula Worksheet A B C D E 1 M J M Cars J Cars Sample Size Sample Mean Sample Std. Dev Est. of Variance Standard Error Confid. Coeff Level of Signif Degr. of Freedom t Value Margin of Error Point Est. of Diff Lower Limit Upper Limit Note: Rows are not shown. Slide 33

34 Hypotheses H : Ha: Hypothesis Tests About m 1 m 2: s 1 and s 2 Unknown m m D m m D H : m m D Ha: m m D H : Ha: m m D m m D Left-tailed Right-tailed Two-tailed Test Statistic t ( x x ) D s n s n Slide 34

35 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-per-gallon performance of J cars? Slide 35

36 Hypothesis Tests About m 1 m 2: s 1 and s 2 Unknown p Value and Critical Value Approaches 1. Develop the hypotheses. H 0 : m 1 - m 2 < 0 H a : m 1 - m 2 > 0 where: m 1 = mean mpg for the population of M cars m 2 = mean mpg for the population of J cars Slide 36

37 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. = Compute the value of the test statistic. t ( x x ) D ( ) s1 s2 (2.56) (1.81) n n Slide 37

38 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: df 2 2 (2.56) (1.81) (2.56) 1 (1.81) Because t = > t.005 = 1.683, the p value < Slide 38

39 Hypothesis Tests About m 1 m 2: s 1 and s 2 Unknown p Value Approach 5. Determine whether to reject H 0. Because p value < =.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?. Slide 39

40 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 = 41, t.05 = Reject H 0 if t > Determine whether to reject H 0. Because > 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?. Slide 40

41 Hypothesis Tests About m 1 m 2: s 1 and s 2 Unknown Excel s t-test: Two Sample Assuming Unequal Variances Tool Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose t-test: Two Sample Assuming Unequal Variances from the list of Analysis Tools continued Slide 41

42 Hypothesis Tests About m 1 m 2: s 1 and s 2 Unknown Excel Dialog Box Slide 42

43 Hypothesis Tests About m 1 m 2: s 1 and s 2 Unknown Excel Value Worksheet A B C D E F z-test: Two-Sample Assuming Unequal Variances 1 Mcar Jcar Mcar Jcar Mean Variance Observations Hypothesized Mean Difference df t Stat P(T<=t) one-tail E t Critical one-tail P(T<=t) two-tail t Critical two-tail Note: Rows are not shown. Slide 43

44 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. Slide 44

45 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. Slide 45

46 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. Slide 46

47 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 Slide 47

48 Inferences About the Difference Between Two Population Means: Matched Samples p Value and Critical Value Approaches 1. Develop the hypotheses. H 0 : m d = 0 H a : m d Let m d = the mean of the difference values for the two delivery services for the population of district offices Slide 48

49 Inferences About the Difference Between Two Population Means: Matched Samples p Value and Critical Value Approaches 2. Specify the level of significance. = Compute the value of the test statistic. d d n i ( ) s d 2 ( di d ) n 1 9 d md t s n d Slide 49

50 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 H 0. Because p value < =.05, we reject H 0. We are at least 95% confident that there is a difference in mean delivery times for the two services? Slide 50

51 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 = Reject H 0 if t > Determine whether to reject H 0. 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? Slide 51

52 Inferences About the Difference Between Two Population Means: Matched Samples Excel s t-test: Paired Two Sample for Means Tool Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose t-test: Paired Two Sample for Means from the list of Analysis Tools continued Slide 52

53 Inferences About the Difference Between Two Population Means: Matched Samples Excel Dialog Box Slide 53

54 Inferences About the Difference Between Two Population Means: Matched Samples Excel Value Worksheet A B C D E F G 1 Office UPX INTEX 2 Seattle t-test: Paired Two Sample for Means 3 L.A Boston UPX INTEX 5 Cleveland Mean N.Y.C Variance Houston Observations Atlanta Pearson Correlation St. Louis 10 8 Hypothesized Mean Difference 0 10 Milwauk. 7 9 df 9 11 Denver t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Slide 54

55 End of Chapter 10 Part A Slide 55