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1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach

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2 2 Slide IS 310 – Business Statistics Inferences on Two Populations In the past, we dealt with one population mean and one population proportion. However, there are situations where two populations are involved dealing with two means. Examples are the following: O We want to compare the mean salaries of male and female graduates (two populations and two means). O We want to compare the mean miles per gallon(MPG) of two comparable automobile makes (two populations and two means)

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3 3 Slide IS 310 – Business Statistics Statistical Inferences About Means and Proportions with Two Populations n Inferences About the Difference Between Two Population Means: 1 and 2 Known Two Population Means: 1 and 2 Known n n Inferences About the Difference Between Two Population Means: 1 and 2 Unknown Two Population Means: 1 and 2 Unknown

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4 4 Slide IS 310 – Business Statistics 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

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5 5 Slide IS 310 – Business Statistics 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.

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6 6 Slide IS 310 – Business Statistics 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

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7 7 Slide IS 310 – Business Statistics n Interval Estimate Interval Estimation of 1 - 2 : 1 and 2 Known where: 1 - is the confidence coefficient 1 - is the confidence coefficient

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8 8 Slide IS 310 – Business Statistics 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.” n Example: Par, Inc.

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9 9 Slide IS 310 – Business Statistics n Example: Par, Inc. 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.

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10 Slide IS 310 – Business Statistics Interval Estimation of 1 - 2 : 1 and 2 Known n Example: Par, Inc. Let us develop a 95% confidence interval estimate Let us develop a 95% confidence interval estimate of the difference between the mean driving distances of the two brands of golf ball.

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11 Slide IS 310 – Business Statistics 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

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12 Slide IS 310 – Business Statistics 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

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13 Slide IS 310 – Business Statistics Interval Estimation of 1 - 2 : 1 and 2 Known 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

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14 Slide IS 310 – Business Statistics Hypothesis Tests About 1 2 : 1 and 2 Known Hypotheses Hypotheses Left-tailedRight-tailedTwo-tailed Test Statistic Test Statistic

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15 Slide IS 310 – Business Statistics n Example: Par, Inc. 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?

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16 Slide IS 310 – Business Statistics 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. p –Value and Critical Value Approaches p –Value and Critical Value Approaches Hypothesis Tests About 1 2 : 1 and 2 Known 2. Specify the level of significance. =.01

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17 Slide IS 310 – Business Statistics 3. Compute the value of the test statistic. Hypothesis Tests About 1 2 : 1 and 2 Known p –Value and Critical Value Approaches p –Value and Critical Value Approaches

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18 Slide IS 310 – Business Statistics p –Value Approach 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.

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19 Slide IS 310 – Business Statistics Hypothesis Tests About 1 2 : 1 and 2 Known 5. Determine whether to reject H 0. Because z = 6.49 > 2.33, we reject H 0. Critical Value Approach Critical Value Approach For =.01, z.01 = 2.33 4. Determine the critical value and rejection rule. Reject H 0 if z > 2.33 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.

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20 Slide IS 310 – Business Statistics Sample Problem Problem # 7 (10-Page 401; 11-Page 414) a. H : µ = µ H : µ > µ 0 1 2 a 1 2 0 1 2 a 1 2 b. Point reduction in the mean duration of games during 2003 = 172 – 166 = 6 minutes = 6 minutes _ _ 2 2 _ _ 2 2 c. Test-statistic, z = [( x - x ) – 0] /√ [ (σ / n ) + (σ / n )] 1 2 1 1 2 2 1 2 1 1 2 2 =(172 – 166)/√[ (144/60 + 144/50)] =(172 – 166)/√[ (144/60 + 144/50)] = 6/2.3 = 2.61 = 6/2.3 = 2.61 Critical z at = 1.645 Reject H 0.05 0 0.05 0 Statistical test supports that the mean duration of games in 2003 is less than that in 2002. p-value = 1 – 0.9955 = 0.0045

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21 Slide IS 310 – Business Statistics 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

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22 Slide IS 310 – Business Statistics 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. use the sample standard deviations s 1 and s 2 use the sample standard deviations s 1 and s 2 as estimates of 1 and 2, and

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23 Slide IS 310 – Business Statistics Interval Estimation of µ - µ 1 2 n (Unknown and ) n 1 2 n Interval estimate n _ _ 2 2 n (x - x ) ± t √ (s /n + s /n ) n 1 2 /2 1 1 2 2 n Degree of freedom = n + n - 2 n 1 2

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24 Slide IS 310 – Business Statistics n Example: Specific Motors 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.

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25 Slide IS 310 – Business Statistics Difference Between Two Population Means: 1 and 2 Unknown n Example: Specific Motors 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

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26 Slide IS 310 – Business Statistics Difference Between Two Population Means: 1 and 2 Unknown Let us develop a 90% confidence Let us develop a 90% confidence interval estimate of the difference between the mpg performances of the two models of automobile. n Example: Specific Motors

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27 Slide IS 310 – Business Statistics 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

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28 Slide IS 310 – Business Statistics Interval Estimate of µ - µ 1 2 n Interval estimate n 2 2 n 29.8 – 27.3 ± t √ (2.56) /24 + (1.81) /28) n 0.1/2 n 2.5 ± 1.676 (0.62) n 2.5 ± 1.04 n 1.46 and 3.54 n We are 90% confident that the difference between the average miles per gallon between the J cars and M cars is between 1.46 and 3.54.

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29 Slide IS 310 – Business Statistics Hypothesis Tests About 1 2 : 1 and 2 Unknown n Hypotheses Left-tailedRight-tailedTwo-tailed n Test Statistic

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30 Slide IS 310 – Business Statistics n Example: Specific Motors Hypothesis Tests About 1 2 : 1 and 2 Unknown Can we conclude, using a 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?

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31 Slide IS 310 – Business Statistics 2. Specify the level of significance. 3. Compute the value of the test statistic. =.05 p –Value and Critical Value Approaches p –Value and Critical Value Approaches Hypothesis Tests About 1 2 : 1 and 2 Unknown

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32 Slide IS 310 – Business Statistics Hypothesis Tests of µ - µ 1 2 n H : µ = µ H : µ > µ n 0 1 2 a 1 2 n Where µ average miles per gallon of M cars n 1 n µ average miles per gallon of J cars n 2 n At = 0.05 with 50 degree of freedom, critical t = 1.676 n Since t-statistic (4.003) is larger than critical t (1.676), we reject the null hypothesis. This means that the average MPG of M cars is not equal to that of J cars

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33 Slide IS 310 – Business Statistics End of Chapter 10 Part A

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