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IS 310 Business Statistics CSU Long Beach

IS 310 Business Statistics CSU Long Beach. 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:

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IS 310 Business Statistics CSU Long Beach

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  1. IS 310 Business Statistics CSU Long Beach

  2. 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)

  3. Statistical Inferences About Meansand Proportions with Two Populations • Inferences About the Difference Between Two Population Means: s1 and s2 Known • Inferences About the Difference Between Two Population Means: s1 and s2 Unknown

  4. Inferences About the Difference BetweenTwo Population Means: s 1 and s 2 Known • Interval Estimation of m1 – m2 • Hypothesis Tests About m1 – m2

  5. 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 . Estimating the Difference BetweenTwo Population Means • Let 1 equal the mean of population 1 and 2 equal the mean of population 2. • The difference between the two population means is 1 - 2. • To estimate 1 - 2, we will select a simple random sample of size n1 from population 1 and a simple random sample of size n2 from population 2.

  6. Sampling Distribution of • Expected Value • Standard Deviation (Standard Error) where: 1 = standard deviation of population 1 2 = standard deviation of population 2 n1 = sample size from population 1 n2 = sample size from population 2

  7. Interval Estimation of 1 - 2:s 1 and s 2 Known • Interval Estimate where: 1 -  is the confidence coefficient

  8. Interval Estimation of 1 - 2:s 1 and s 2 Known • 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.

  9. Interval Estimation of 1 - 2:s 1 and s 2 Known • Example: Par, Inc. Sample #1 Par, Inc. Sample #2 Rap, Ltd. Sample Size 120 balls 80 balls Sample Mean 275 yards 258 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.

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

  11. Population 1 Par, Inc. Golf Balls m1 = mean driving distance of Par golf balls Population 2 Rap, Ltd. Golf Balls m2 = mean driving distance of Rap golf balls Simple random sample of n1 Par golf balls x1 = sample mean distance for the Par golf balls Simple random sample of n2 Rap golf balls x2 = sample mean distance for the Rap golf balls x1 - x2 = Point Estimate of m1 –m2 Estimating the Difference BetweenTwo Population Means m1 –m2= difference between the mean distances

  12. Point Estimate of 1 - 2 Point estimate of 1-2 = = 275 - 258 = 17 yards where: 1 = mean distance for the population of Par, Inc. golf balls 2 = mean distance for the population of Rap, Ltd. golf balls

  13. Interval Estimation of 1 - 2:1 and 2 Known 17 + 5.14 or 11.86 yards to 22.14 yards 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.

  14. Hypothesis Tests About m 1-m 2:s 1 and s 2 Known • Hypotheses Left-tailed Right-tailed Two-tailed • Test Statistic

  15. Hypothesis Tests About m 1-m 2:s 1 and s 2 Known • Example: Par, Inc. Can we conclude, using a = .01, that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls?

  16. Hypothesis Tests About m 1-m 2:s 1 and s 2 Known • p –Value and Critical Value Approaches 1. Develop the hypotheses. H0: 1 - 2< 0 Ha: 1 - 2 > 0 where: 1 = mean distance for the population of Par, Inc. golf balls 2 = mean distance for the population of Rap, Ltd. golf balls a = .01 2. Specify the level of significance.

  17. 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.

  18. 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 < .0001. 5. Determine whether to reject H0. Because p–value <a = .01, we reject H0. 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.

  19. 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 a = .01, z.01 = 2.33 Reject H0 if z> 2.33 5. Determine whether to reject H0. Because z = 6.49 > 2.33, we reject H0. 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.

  20. Sample Problem Problem # 7 (10-Page 401; 11-Page 414) a. H : µ = µ H : µ > µ 0 1 2 a 1 2 • Point reduction in the mean duration of games during 2003 = 172 – 166 = 6 minutes _ _ 2 2 c. Test-statistic, z = [( x - x ) – 0] /√ [ (σ / n ) + (σ / n )] 1 2 1 1 2 2 =(172 – 166)/√[ (144/60 + 144/50)] = 6/2.3 = 2.61 Critical z at  = 1.645 Reject H 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

  21. Inferences About the Difference BetweenTwo Population Means: s 1 and s 2 Unknown • Interval Estimation of m1 – m2 • Hypothesis Tests About m1 – m2

  22. Interval Estimation of 1 - 2:s 1 and s 2 Unknown When s 1 and s 2 are unknown, we will: • use the sample standard deviations s1 and s2 • as estimates of s 1 and s 2 , and • replace za/2 with ta/2.

  23. Interval Estimation of µ - µ 1 2 • (Unknown  and  ) • 1 2 • Interval estimate • _ _ 2 2 • (x - x ) ± t √ (s /n + s /n ) • 1 2 /2 1 1 2 2 • Degree of freedom = n + n - 2 • 1 2

  24. 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-per-gallon (mpg) performance. The sample statistics are shown on the next slide.

  25. 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 Sample Size 29.8 mpg 27.3 mpg Sample Mean 2.56 mpg 1.81 mpg Sample Std. Dev.

  26. Difference Between Two Population Means: s 1 and s 2 Unknown • Example: Specific Motors Let us develop a 90% confidence interval estimate of the difference between the mpg performances of the two models of automobile.

  27. Point Estimate of m 1-m 2 Point estimate of 1-2 = = 29.8 - 27.3 = 2.5 mpg where: 1 = mean miles-per-gallon for the population of M cars 2 = mean miles-per-gallon for the population of J cars

  28. Interval Estimate of µ - µ 1 2 • Interval estimate • 2 2 • 29.8 – 27.3 ± t √ (2.56) /24 + (1.81) /28) • 0.1/2 • 2.5 ± 1.676 (0.62) • 2.5 ± 1.04 • 1.46 and 3.54 • 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.

  29. Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown • Hypotheses Left-tailed Right-tailed Two-tailed • Test Statistic

  30. 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?

  31. Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown • p –Value and Critical Value Approaches a = .05 2. Specify the level of significance. 3. Compute the value of the test statistic.

  32. Hypothesis Tests of µ - µ 1 2 • H : µ = µ H : µ > µ • 0 1 2 a 1 2 • Where µ average miles per gallon of M cars • 1 • µ average miles per gallon of J cars • 2 • At  = 0.05 with 50 degree of freedom, critical t = 1.676 • 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

  33. End of Chapter 10Part A

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