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

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.

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

Dependent Samples Tests Means of 2 Related Populations di = xi - yi 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 di = xi - yi Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Mean Difference The ith paired difference is di , where di = xi - yi Dependent samples di = xi - yi The point estimate for the population mean paired difference is d : The sample standard deviation is: n is the number of matched pairs in the sample Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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

Confidence Interval for Mean Difference (continued) The margin of error is tn-1,/2 is the value from the Student’s t distribution with (n – 1) degrees of freedom for which Dependent samples Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Paired Samples Example Six people sign up for a weight loss program. You collect the following data:  Weight: Person Before (x) After (y) Difference, di 1 136 125 11 2 205 195 10 3 157 150 7 4 138 140 - 2 5 175 165 10 6 166 160 6 42 di d = n = 7.0 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Paired Samples Example (continued) For a 95% confidence level, the appropriate t value is tn-1,/2 = t5,.025 = 2.571 The 95% confidence interval for the difference between means, μd , is 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.

Difference Between Two Means Goal: Form a confidence interval for the difference between two population means, μx – μy Population means, independent samples 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: x – y Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Difference Between Two Means (continued) Population means, independent samples σx2 and σy2 known Confidence interval uses z/2 σx2 and σy2 unknown σx2 and σy2 assumed equal Confidence interval uses a value from the Student’s t distribution σx2 and σy2 assumed unequal Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Population means, independent samples σx2 and σy2 Known Population means, independent samples Assumptions: Samples are randomly and independently drawn both population distributions are normal Population variances are known * σx2 and σy2 known σx2 and σy2 unknown Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Population means, independent samples σx2 and σy2 Known (continued) When σx and σy are known and both populations are normal, the variance of X – Y is Population means, independent samples * σx2 and σy2 known …and the random variable has a standard normal distribution σx2 and σy2 unknown Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Confidence Interval, σx2 and σy2 Known Population means, independent samples * σx2 and σy2 known The confidence interval for μx – μy is: σx2 and σy2 unknown Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

σx2 and σy2 Unknown, Assumed Equal Assumptions: Samples are randomly and independently drawn Populations are normally distributed Population variances are unknown but assumed equal Population means, independent samples σx2 and σy2 known σx2 and σy2 unknown * σx2 and σy2 assumed equal σx2 and σy2 assumed unequal Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

σx2 and σy2 Unknown, Assumed Equal (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 (nx + ny – 2) degrees of freedom Population means, independent samples σx2 and σy2 known σx2 and σy2 unknown * σx2 and σy2 assumed equal σx2 and σy2 assumed unequal Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

σx2 and σy2 Unknown, Assumed Equal (continued) Population means, independent samples The pooled variance is σx2 and σy2 known σx2 and σy2 unknown * σx2 and σy2 assumed equal σx2 and σy2 assumed unequal Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Confidence Interval, σx2 and σy2 Unknown, Equal * σx2 and σy2 assumed equal The confidence interval for μ1 – μ2 is: σx2 and σy2 assumed unequal Where Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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): CPUx CPUy Number Tested 17 14 Sample mean 3004 2538 Sample std dev 74 56 Assume both populations are normal with equal variances, and use 95% confidence Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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.

Calculating the Confidence Limits The 95% confidence interval is We are 95% confident that the mean difference in CPU speed is between 416.69 and 515.31 Mhz. Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

σx2 and σy2 Unknown, Assumed Unequal Assumptions: Samples are randomly and independently drawn Populations are normally distributed Population variances are unknown and assumed unequal Population means, independent samples σx2 and σy2 known σx2 and σy2 unknown σx2 and σy2 assumed equal * σx2 and σy2 assumed unequal Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

σx2 and σy2 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 Population means, independent samples σx2 and σy2 known σx2 and σy2 unknown σx2 and σy2 assumed equal * σx2 and σy2 assumed unequal Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Confidence Interval, σx2 and σy2 Unknown, Unequal σx2 and σy2 assumed equal The confidence interval for μ1 – μ2 is: * σx2 and σy2 assumed unequal Where Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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

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

Confidence Interval for Two Population Proportions The confidence limits for Px – Py are: Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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.

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

Example: Two Population Proportions (continued) The confidence limits are: so the confidence interval is -0.3465 < Px – Py < -0.0135 Since this interval does not contain zero we are 90% confident that the two proportions are not equal Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

Confidence Intervals for the Population Variance Goal: Form a confidence interval for the population variance, σ2 The confidence interval is based on the sample variance, s2 Assumed: the population is normally distributed Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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

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

Example You are testing the speed of a computer processor. You collect the following data (in Mhz): CPUx Sample size 17 Sample mean 3004 Sample std dev 74 Assume the population is normal. Determine the 95% confidence interval for σx2 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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 0.025 in each tail: probability α/2 = .025 probability α/2 = .025 216 216 = 6.91 216 = 28.85 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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 112.6 Mhz Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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

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

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

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.

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

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

Sample Size Determination (continued) 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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

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

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

Sample Size Determination (continued) 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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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.

Required Sample Size Example (continued) Solution: For 95% confidence, use z0.025 = 1.96 ME = 0.03 Estimate P(1 – P) = 0.25 So use n = 1068 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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

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 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.