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

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

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

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

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

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

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 Student’s t distribution with (n – 1) degrees of freedom for which Confidence Interval for Mean Difference (continued) Dependent samples

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

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

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:

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 Student’s 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)

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

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

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

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

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

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

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

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

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:

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.

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

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

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

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)

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

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:

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

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

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)

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

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

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)

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

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 =.025  2 16 =  2 16 = 6.91 probability α/2 =.025

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

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

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

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

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

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

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

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

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

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

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

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

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?

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)

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

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