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1 Chap 9-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 9 Estimation: Additional Topics Statistics for Business and Economics 6 th Edition

2 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-2 Chapter Goals After completing this chapter, you should be able to: Form confidence intervals for the mean difference from dependent samples Form confidence intervals for the difference between two independent population means ( standard deviations known or unknown) Compute confidence interval limits for the difference between two independent population proportions Create confidence intervals for a population variance Find chi-square values from the chi-square distribution table Determine the required sample size to estimate a mean or proportion within a specified margin of error

3 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-3 Estimation: Additional Topics Chapter Topics Population Means, Independent Samples Population Means, Dependent Samples Population Variance Group 1 vs. independent Group 2 Same group before vs. after treatment Variance of a normal distribution Examples: Population Proportions Proportion 1 vs. Proportion 2

4 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-4 Dependent Samples Tests Means of 2 Related Populations Paired or matched samples Repeated measures (before/after) Use difference between paired values: Eliminates Variation Among Subjects Assumptions: Both Populations Are Normally Distributed Dependent samples d i = x i - y i

5 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-5 Mean Difference The i th paired difference is d i, where d i = x i - y i The point estimate for the population mean paired difference is d : n is the number of matched pairs in the sample The sample standard deviation is: Dependent samples

6 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-6 Confidence Interval for Mean Difference The confidence interval for difference between population means, μ d, is Where n = the sample size (number of matched pairs in the paired sample) Dependent samples

7 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-7 The margin of error is t n-1, /2 is the value from the Students t distribution with (n – 1) degrees of freedom for which Confidence Interval for Mean Difference (continued) Dependent samples

8 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-8 Six people sign up for a weight loss program. You collect the following data: Paired Samples Example Weight: Person Before (x) After (y) Difference, d i d = didi n = 7.0

9 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-9 For a 95% confidence level, the appropriate t value is t n-1, /2 = t 5,.025 = The 95% confidence interval for the difference between means, μ d, is Paired Samples Example (continued) Since this interval contains zero, we cannot be 95% confident, given this limited data, that the weight loss program helps people lose weight

10 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-10 Difference Between Two Means Population means, independent samples Goal: Form a confidence interval for the difference between two population means, μ x – μ y x – y Different data sources Unrelated Independent Sample selected from one population has no effect on the sample selected from the other population The point estimate is the difference between the two sample means:

11 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-11 Difference Between Two Means Population means, independent samples Confidence interval uses z /2 Confidence interval uses a value from the Students t distribution σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal (continued)

12 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-12 Population means, independent samples σ x 2 and σ y 2 Known Assumptions: Samples are randomly and independently drawn both population distributions are normal Population variances are known * σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown

13 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-13 Population means, independent samples …and the random variable has a standard normal distribution When σ x and σ y are known and both populations are normal, the variance of X – Y is (continued) * σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 Known

14 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-14 Population means, independent samples The confidence interval for μ x – μ y is: Confidence Interval, σ x 2 and σ y 2 Known * σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown

15 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-15 Population means, independent samples σ x 2 and σ y 2 Unknown, Assumed Equal Assumptions: Samples are randomly and independently drawn Populations are normally distributed Population variances are unknown but assumed equal * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal

16 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-16 Population means, independent samples (continued) Forming interval estimates: The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ use a t value with (n x + n y – 2) degrees of freedom * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal σ x 2 and σ y 2 Unknown, Assumed Equal

17 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-17 Population means, independent samples The pooled variance is (continued) * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal σ x 2 and σ y 2 Unknown, Assumed Equal

18 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-18 The confidence interval for μ 1 – μ 2 is: Where * Confidence Interval, σ x 2 and σ y 2 Unknown, Equal σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal

19 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-19 Pooled Variance Example You are testing two computer processors for speed. Form a confidence interval for the difference in CPU speed. You collect the following speed data (in Mhz): CPU x CPU y Number Tested Sample mean Sample std dev Assume both populations are normal with equal variances, and use 95% confidence

20 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-20 Calculating the Pooled Variance The pooled variance is: The t value for a 95% confidence interval is:

21 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-21 Calculating the Confidence Limits The 95% confidence interval is We are 95% confident that the mean difference in CPU speed is between and Mhz.

22 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-22 Population means, independent samples σ x 2 and σ y 2 Unknown, Assumed Unequal Assumptions: Samples are randomly and independently drawn Populations are normally distributed Population variances are unknown and assumed unequal * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal

23 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-23 Population means, independent samples σ x 2 and σ y 2 Unknown, Assumed Unequal (continued) Forming interval estimates: The population variances are assumed unequal, so a pooled variance is not appropriate use a t value with degrees of freedom, where σ x 2 and σ y 2 known σ x 2 and σ y 2 unknown * σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 assumed unequal

24 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-24 The confidence interval for μ 1 – μ 2 is: * Confidence Interval, σ x 2 and σ y 2 Unknown, Unequal σ x 2 and σ y 2 assumed equal σ x 2 and σ y 2 unknown σ x 2 and σ y 2 assumed unequal Where

25 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-25 Two Population Proportions Goal: Form a confidence interval for the difference between two population proportions, P x – P y The point estimate for the difference is Population proportions Assumptions: Both sample sizes are large (generally at least 40 observations in each sample)

26 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-26 Two Population Proportions Population proportions (continued) The random variable is approximately normally distributed

27 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-27 Confidence Interval for Two Population Proportions Population proportions The confidence limits for P x – P y are:

28 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-28 Example: Two Population Proportions Form a 90% confidence interval for the difference between the proportion of men and the proportion of women who have college degrees. In a random sample, 26 of 50 men and 28 of 40 women had an earned college degree

29 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-29 Example: Two Population Proportions Men: Women: (continued) For 90% confidence, Z /2 = 1.645

30 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-30 Example: Two Population Proportions The confidence limits are: so the confidence interval is < P x – P y < Since this interval does not contain zero we are 90% confident that the two proportions are not equal (continued)

31 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-31 Confidence Intervals for the Population Variance Population Variance Goal: Form a confidence interval for the population variance, σ 2 The confidence interval is based on the sample variance, s 2 Assumed: the population is normally distributed

32 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-32 Confidence Intervals for the Population Variance Population Variance The random variable follows a chi-square distribution with (n – 1) degrees of freedom (continued) The chi-square value denotes the number for which

33 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-33 Confidence Intervals for the Population Variance Population Variance The (1 - )% confidence interval for the population variance is (continued)

34 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-34 Example You are testing the speed of a computer processor. You collect the following data (in Mhz): CPU x Sample size 17 Sample mean 3004 Sample std dev 74 Assume the population is normal. Determine the 95% confidence interval for σ x 2

35 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-35 Finding the Chi-square Values n = 17 so the chi-square distribution has (n – 1) = 16 degrees of freedom = 0.05, so use the the chi-square values with area in each tail: probability α/2 = = = 6.91 probability α/2 =.025

36 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-36 Calculating the Confidence Limits The 95% confidence interval is Converting to standard deviation, we are 95% confident that the population standard deviation of CPU speed is between 55.1 and Mhz

37 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-37 Sample PHStat Output

38 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-38 Sample PHStat Output Input Output (continued)

39 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-39 Sample Size Determination For the Mean Determining Sample Size For the Proportion

40 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-40 Margin of Error The required sample size can be found to reach a desired margin of error (ME) with a specified level of confidence (1 - ) The margin of error is also called sampling error the amount of imprecision in the estimate of the population parameter the amount added and subtracted to the point estimate to form the confidence interval

41 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-41 For the Mean Determining Sample Size Margin of Error (sampling error) Sample Size Determination

42 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-42 For the Mean Determining Sample Size (continued) Now solve for n to get Sample Size Determination

43 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-43 To determine the required sample size for the mean, you must know: The desired level of confidence (1 - ), which determines the z /2 value The acceptable margin of error (sampling error), ME The standard deviation, σ (continued) Sample Size Determination

44 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-44 Required Sample Size Example If = 45, what sample size is needed to estimate the mean within ± 5 with 90% confidence? (Always round up) So the required sample size is n = 220

45 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-45 Determining Sample Size For the Proportion Margin of Error (sampling error) Sample Size Determination

46 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-46 Determining Sample Size For the Proportion Substitute 0.25 for and solve for n to get (continued) Sample Size Determination cannot be larger than 0.25, when = 0.5

47 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-47 The sample and population proportions, and P, are generally not known (since no sample has been taken yet) P(1 – P) = 0.25 generates the largest possible margin of error (so guarantees that the resulting sample size will meet the desired level of confidence) To determine the required sample size for the proportion, you must know: The desired level of confidence (1 - ), which determines the critical z /2 value The acceptable sampling error (margin of error), ME Estimate P(1 – P) = 0.25 (continued) Sample Size Determination

48 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-48 Required Sample Size Example How large a sample would be necessary to estimate the true proportion defective in a large population within ±3%, with 95% confidence?

49 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-49 Required Sample Size Example Solution: For 95% confidence, use z = 1.96 ME = 0.03 Estimate P(1 – P) = 0.25 So use n = 1068 (continued)

50 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-50 PHStat Sample Size Options

51 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 9-51 Chapter Summary Compared two dependent samples (paired samples) Formed confidence intervals for the paired difference Compared two independent samples Formed confidence intervals for the difference between two means, population variance known, using z Formed confidence intervals for the differences between two means, population variance unknown, using t Formed confidence intervals for the differences between two population proportions Formed confidence intervals for the population variance using the chi-square distribution Determined required sample size to meet confidence and margin of error requirements

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