Statistics for Business and Economics 7 th Edition
Statistics for Business and Economics 7 th Edition Chapter 8 Estimation: Additional Topics Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -1
Chapter Goals After completing this chapter, you should be able to: n n Form confidence intervals for the difference between two means 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 Determine the required sample size to estimate a mean or proportion within a specified margin of error Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -2
Estimation: Additional Topics Chapter Topics Confidence Intervals Population Means, Dependent Samples Population Means, Independent Samples Population Proportions Examples: Same group before vs. after treatment Group 1 vs. independent Group 2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Proportion 1 vs. Proportion 2 Sample Size Determination Large Populations Finite Populations Ch. 8 -3
8. 1 Dependent Samples Tests Means of 2 Related Populations Dependent samples n n n Paired or matched samples Repeated measures (before/after) Use difference between paired values: di = x i - y i n n Eliminates Variation Among Subjects Assumptions: n Both Populations Are Normally Distributed Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -4
Mean Difference Dependent samples The ith paired difference is di , where di = x i - y i 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -5
Confidence Interval for Mean Difference Dependent samples The confidence interval for difference between population means, μd , is Where n = the sample size (number of matched pairs in the paired sample) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -6
Confidence Interval for Mean Difference (continued) Dependent samples n n 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -7
Paired Samples Example Dependent samples Person 1 2 3 4 5 6 Six people sign up for a weight loss program. You collect the following data: n Weight: Before (x) After (y) 136 205 157 138 175 166 125 195 150 140 165 160 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Difference, di 11 10 7 -2 10 6 42 di d = n = 7. 0 Ch. 8 -8
Paired Samples Example (continued) Dependent samples n n For a 95% confidence level, the appropriate t value is tn -1, /2 = t 5, . 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -9
Difference Between Two Means: Independent Samples 8. 2 Population means, independent samples n Different data sources n Unrelated n Independent n n Goal: Form a confidence interval for the difference between two population means, μx – μy 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -10
Difference Between Two Means: Independent Samples (continued) Population means, independent samples σx 2 and σy 2 known Confidence interval uses z /2 σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal σx 2 and σy 2 assumed unequal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Confidence interval uses a value from the Student’s t distribution Ch. 8 -11
σx 2 and σy 2 Known Population means, independent samples σx 2 and σy 2 known Assumptions: * σx 2 and σy 2 unknown § Samples are randomly and independently drawn § both population distributions are normal § Population variances are known Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -12
σx 2 and σy 2 Known (continued) When σx and σy are known and both populations are normal, the variance of X – Y is Population means, independent samples σx 2 and σy 2 known σx 2 and σy 2 unknown * …and the random variable has a standard normal distribution Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -13
Confidence Interval, σx 2 and σy 2 Known Population means, independent samples σx 2 and σy 2 known * σx 2 and σy 2 unknown Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall The confidence interval for μx – μy is: Ch. 8 -14
σx 2 and σy 2 Unknown, Assumed Equal Assumptions: Population means, independent samples § Samples are randomly and independently drawn σx 2 and σy 2 known § Populations are normally distributed σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal * § Population variances are unknown but assumed equal σx 2 and σy 2 assumed unequal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -15
σx 2 and σy 2 Unknown, Assumed Equal (continued) Forming interval estimates: Population means, independent samples § The population variances are assumed equal, so use the two sample standard deviations and pool them to estimate σ σx 2 and σy 2 known σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal * σx 2 and σy 2 assumed unequal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall § use a t value with (nx + ny – 2) degrees of freedom Ch. 8 -16
σx 2 and σy 2 Unknown, Assumed Equal (continued) Population means, independent samples The pooled variance is σx 2 and σy 2 known σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal * σx 2 and σy 2 assumed unequal Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -17
Confidence Interval, σx 2 and σy 2 Unknown, Equal σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal σx 2 and σy 2 assumed unequal * The confidence interval for μ 1 – μ 2 is: Where Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -18
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): Number Tested Sample mean Sample std dev CPUx 17 3004 74 CPUy 14 2538 56 Assume both populations are normal with equal variances, and use 95% confidence Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -19
Calculating the Pooled Variance The pooled variance is: The t value for a 95% confidence interval is: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -20
Calculating the Confidence Limits n The 95% confidence interval is We are 95% confident that the mean difference in CPU speed is between 416. 69 and 515. 31 Mhz. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -21
σx 2 and σy 2 Unknown, Assumed Unequal Assumptions: Population means, independent samples § Samples are randomly and independently drawn σx 2 and σy 2 known § Populations are normally distributed σx 2 and σy 2 unknown § Population variances are unknown and assumed unequal σx 2 and σy 2 assumed unequal * Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -22
σx 2 and σy 2 Unknown, Assumed Unequal (continued) Forming interval estimates: Population means, independent samples § The population variances are assumed unequal, so a pooled variance is not appropriate σx 2 and σy 2 known § use a t value with degrees of freedom, where σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal σx 2 and σy 2 assumed unequal * Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -23
Confidence Interval, σx 2 and σy 2 Unknown, Unequal σx 2 and σy 2 unknown σx 2 and σy 2 assumed equal σx 2 and σy 2 assumed unequal * The confidence interval for μ 1 – μ 2 is: Where Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -24
8. 3 Population proportions Two Population Proportions Goal: Form a confidence interval for the difference between two population proportions, Px – Py Assumptions: Both sample sizes are large (generally at least 40 observations in each sample) The point estimate for the difference is Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -25
Two Population Proportions (continued) Population proportions n The random variable is approximately normally distributed Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -26
Confidence Interval for Two Population Proportions Population proportions The confidence limits for Px – Py are: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -27
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. n In a random sample, 26 of 50 men and 28 of 40 women had an earned college degree Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -28
Example: Two Population Proportions (continued) Men: Women: For 90% confidence, Z /2 = 1. 645 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -29
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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -30
8. 4 Sample Size Determination Determining Sample Size Large Populations For the Mean For the Proportion Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Finite Populations For the Mean For the Proportion Ch. 8 -31
Margin of Error n n 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 n n 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -32
Sample Size Determination Large Populations For the Mean Margin of Error (sampling error) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -33
Sample Size Determination Large Populations (continued) For the Mean Now solve for n to get Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -34
Sample Size Determination (continued) n To determine the required sample size for the mean, you must know: n The desired level of confidence (1 - ), which determines the z /2 value n The acceptable margin of error (sampling error), ME n The population standard deviation, σ Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -35
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) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -36
Sample Size Determination: Population Proportion Large Populations For the Proportion Margin of Error (sampling error) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -37
Sample Size Determination: Population Proportion (continued) Large Populations For the Proportion cannot be larger than 0. 25, when = 0. 5 Substitute 0. 25 for and solve for n to get Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -38
Sample Size Determination: Population Proportion (continued) n n n 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: n The desired level of confidence (1 - ), which determines the critical z /2 value n The acceptable sampling error (margin of error), ME n Estimate P(1 – P) = 0. 25 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -39
Required Sample Size Example: Population Proportion How large a sample would be necessary to estimate the true proportion defective in a large population within ± 3%, with 95% confidence? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -40
Required Sample Size Example (continued) Solution: For 95% confidence, use z 0. 025 = 1. 96 ME = 0. 03 Estimate P(1 – P) = 0. 25 So use n = 1068 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -41
8. 5 Sample Size Determination: Finite Populations For the Mean A finite population correction factor is added: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 1. Calculate the required sample size n 0 using the prior formula: 2. Then adjust for the finite population: Ch. 8 -42
Sample Size Determination: Finite Populations For the Proportion 1. Solve for n: 2. The largest possible value for this expression (if P = 0. 25) is: A finite population correction factor is added: 3. A 95% confidence interval will extend ± 1. 96 from the sample proportion Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -43
Example: Sample Size to Estimate Population Proportion (continued) How large a sample would be necessary to estimate within ± 5% the true proportion of college graduates in a population of 850 people with 95% confidence? Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -44
Required Sample Size Example (continued) Solution: § For 95% confidence, use z 0. 025 = 1. 96 § ME = 0. 05 So use n = 265 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -45
Chapter Summary n Compared two dependent samples (paired samples) n n Compared two independent samples n n n Formed confidence intervals for the paired difference 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 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 8 -46
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