Chapter 10 Comparisons Involving Means Part A Estimation of the Difference between the Means of Two Populations: Independent Samples Hypothesis Tests about the Difference between the Means of Two Populations: Independent Samples

Estimation of the Difference Between the Means of Two Populations: Independent Samples Point Estimator of the Difference between the Means of Two Populations Sampling Distribution of Interval Estimate of      Large-Sample Case Interval Estimate of      Small-Sample Case

Point Estimator of the Difference Between the Means of Two Populations 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.

n Expected Value Sampling Distribution of n Standard Deviation 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

Interval Estimate with  1 and  2 Known Interval Estimate of  1 -  2 : Large-Sample Case (n 1 > 30 and n 2 > 30) where: 1 -  is the confidence coefficient 1 -  is the confidence coefficient

Interval Estimate with  1 and  2 Unknown Interval Estimate with  1 and  2 Unknown Interval Estimate of  1 -  2 : Large-Sample Case ( n 1 > 30 and n 2 > 30) where:

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. Interval Estimate of  1 -  2 : Large-Sample Case ( n 1 > 30 and n 2 > 30)

Example: Par, Inc. Interval Estimate of  1 -  2 : Large-Sample Case ( n 1 > 30 and n 2 > 30) Sample Size Sample Mean Sample Std. Dev. Sample #1 Par, Inc. Sample #2 Rap, Ltd. 120 balls 80 balls 120 balls 80 balls 235 yards 218 yards 15 yards 20 yards

Point Estimator of the Difference Between the Means of Two Populations 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 m 1 –  2 = difference between the mean distances the mean distances Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for sample of Par golf ball Simple random sample Simple random sample of n 1 Par golf balls of n 1 Par golf balls x 1 = sample mean distance for sample of Par golf ball Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for sample of Rap golf ball Simple random sample Simple random sample of n 2 Rap golf balls of n 2 Rap golf balls x 2 = sample mean distance for sample of Rap golf ball x 1 - x 2 = Point Estimate of m 1 –  2

Point Estimate of the Difference Between Two Population Means 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 = 235  218 = 17 yards

Substituting the sample standard deviations for the population standard deviation: 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case,  1 and  2 Unknown 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 to yards or yards to yards

Using Excel to Develop an Interval Estimate of  1 –  2 : Large-Sample Case n Formula Worksheet Note: Rows are not shown.

n Value Worksheet Using Excel to Develop an Interval Estimate of  1 –  2 : Large-Sample Case Note: Rows are not shown.

Interval Estimate of  1 -  2 : Small-Sample Case (n 1 < 30 and/or n 2 < 30) where: Interval Estimate with Interval Estimate with

Interval Estimate of  1 -  2 : Small-Sample Case (n 1 < 30 and/or n 2 < 30) Interval Estimate with  2 Unknown where:

n 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 miles-per-gallon (mpg) performance. The sample statistics are shown on the next slide. next slide. Difference Between Two Population Means: Small Sample Case

Difference Between Two Population Means: Small Sample Case n Example: Specific Motors Sample Size Sample Mean Sample Std. Dev. Sample #1 M Cars Sample #2 J Cars 12 cars 8 cars 12 cars 8 cars 29.8 mpg 27.3 mpg 2.56 mpg 1.81 mpg

Point estimate of  1   2 = Point Estimate of the Difference Between Two Population Means 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 = = 2.5 mpg

We will make the following assumptions: 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case The variance in the miles per gallon rating The variance in the miles per gallon rating is the same for both the M car and the J car. is the same for both the M car and the J car. The miles per gallon rating is normally The miles per gallon rating is normally distributed for both the M car and the J car. distributed for both the M car and the J car.

95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case We will use a weighted average of the two sample We will use a weighted average of the two sample variances as the pooled estimator of  2.

or.3 to 4.7 miles per gallon 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case n Using the t distribution with n 1 + n = 18 degrees of freedom, the appropriate t value is t.025 = of freedom, the appropriate t value is t.025 = We are 95% confident that the difference between We are 95% confident that the difference between the mean mpg ratings of the two car types is.3 to 4.7 the mean mpg ratings of the two car types is.3 to 4.7 mpg (with the M car having the higher mpg). mpg (with the M car having the higher mpg).

n Formula Worksheet Using Excel to Develop an Interval Estimate of  1 –  2 : Small-Sample

n Value Worksheet Using Excel to Develop an Interval Estimate of  1 –  2 : Small-Sample

Hypothesis Tests About the Difference between the Means of Two Populations: Independent Samples Hypotheses Test Statistic Large-Sample Small-Sample

n Example: Par, Inc. Recall that Par, Inc. has Recall that Par, Inc. has developed a new golf ball that was designed to provide “extra distance.” 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. Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Large-Sample Case

n Example: Par, Inc. Can we conclude, using  =.01, 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? Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Large-Sample Case Sample Size Sample Mean Sample Std. Dev. Sample #1 Par, Inc. Sample #2 Rap, Ltd. 120 balls 80 balls 120 balls 80 balls 235 yards 218 yards 15 yards 20 yards

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. Determine the hypotheses. Using the Test Statistic Using the Test Statistic Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Large-Sample Case

Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Large-Sample Case 2. Specify the level of significance. 3. Select the test statistic.  = State the rejection rule. Reject H 0 if z > 2.33 Using the Test Statistic Using the Test Statistic

Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Large-Sample Case 5. Compute the value of the test statistic. Using the Test Statistic Using the Test Statistic

Hypothesis Tests About the Difference Between the Means of Two Populations: Independent Samples, Large-Sample Case 6. Determine whether to 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. z = 6.49 > z.01 = 2.33, so we reject H 0. Using the Test Statistic Using the Test Statistic

Using Excel to Conduct a Hypothesis Test about  1 –  2 : Large Sample Case n Excel’s “ z -Test: Two Sample for Means” Tool Step 1 Select the Tools menu Step 2 Choose the Data Analysis option Step 3 Choose z -Test: Two Sample for Means from the list of Analysis Tools from the list of Analysis Tools … continued

n Excel’s “ z -Test: Two Sample for Means” Tool Using Excel to Conduct a Hypothesis Test about  1 –  2 : Large Sample Case Step 4 When the z-Test: Two Sample for Means dialog box appears: dialog box appears: … continued Enter A1:A121 in the Variable 1 Range box Enter B1:B81 in the Variable 2 Range box Type 0 in the Hypothesized Mean Difference box Difference box Type 225 in the Variable 1 Variance (known) box (known) box Type 400 in the Variable 2 Variance (known) box (known) box

n Excel’s “ z -Test: Two Sample for Means” Tool Using Excel to Conduct a Hypothesis Test about  1 –  2 : Large Sample Case Click OK (Any upper left-hand corner cell indicating where the output is to begin may be entered) Enter D4 in the Output Range box Select Output Range Type.01 in the Alpha box Select Labels Step 4 (continued)

Using Excel to Conduct a Hypothesis Test about  1 –  2 : Large Sample Case

n Value Worksheet Using Excel to Conduct a Hypothesis Test about  1 –  2 : Large Sample Case Note: Rows are not shown.

Using the p  Value Using the p  Value 4. Compute the value of the test statistic. 5. Compute the p –value. The Excel worksheet states p -value = 4.501E Determine whether to reject H 0. Because p –value <  =.01, we reject H 0. The Excel worksheet states z = 6.48 Using Excel to Conduct a Hypothesis Test about  1 –  2 : Large Sample Case