Statistics for Business and Economics 7th Edition Chapter 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: 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 vs independent Group Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Proportion vs Proportion Sample Size Determination Large Populations Finite Populations Ch 8-3 8.1 Dependent Samples Tests Means of Related Populations Dependent samples Paired or matched samples Repeated measures (before/after) Use difference between paired values: di = x i - y i Eliminates Variation Among Subjects Assumptions: Both Populations Are Normally Distributed Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 8-4 Mean Difference The ith paired difference is di , where Dependent samples di = x i - y i The point estimate for the population mean paired difference is d : The sample standard deviation is: n d= ∑d i =1 i n n Sd = (d − d ) ∑ i i=1 n −1 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 d − t n−1,α/2 Sd Sd < μd < d + t n−1,α/2 n n 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 The margin of error is ME = t n−1,α/2 sd n tn-1,α/2 is the value from the Student’s t distribution with (n – 1) degrees of freedom for which α P(t n−1 > t n−1,α/2 ) = Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 8-7 Paired Samples Example Six people sign up for a weight loss program You collect the following data: Dependent samples Person 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 -2 10 42 Σ di d = n = 7.0 Sd = (d − d ) ∑ i n −1 = 4.82 Ch 8-8 Paired Samples Example (continued) Dependent samples For a 95% confidence level, the appropriate t value is tn-1,α/2 = t5,.025 = 2.571 The 95% confidence interval for the difference between means, μd , is d − t n−1,α/2 − (2.571) Sd S < μd < d + t n−1,α/2 d n n 4.82 4.82 < μd < + (2.571) 6 − 1.94 < μd < 12.06 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 Different data sources Unrelated Independent 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 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 - α) 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 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) x ± z α/2 σ n Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall ME = z α/2 σ n Ch 8-33 Sample Size Determination Large Populations (continued) For the Mean ME = z α/2 σ n Now solve for n to get Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall α/2 z σ n= ME Ch 8-34 Sample Size Determination (continued) To determine the required sample size for the mean, you must know: The desired level of confidence (1 - α), which determines the zα/2 value The acceptable margin of error (sampling error), ME The population standard deviation, σ Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 8-35 Required Sample Size Example If σ = 45, what sample size is needed to estimate the mean within ± with 90% confidence? α/2 2 z σ (1.645) (45) n= = = 219.19 2 ME 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 pˆ ± z α/2 pˆ (1− pˆ ) n ME = z α/2 pˆ (1− pˆ ) n 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 ME = z α/2 pˆ (1− pˆ ) n pˆ (1− pˆ ) cannot be larger than 0.25, when pˆ = 0.5 Substitute 0.25 for pˆ (1− pˆ ) and solve for n to get Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 0.25 z n= ME α/2 Ch 8-38 Sample Size Determination: Population Proportion (continued) The sample and population proportions, pˆ 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 - α), which determines the critical zα/2 value The acceptable sampling error (margin of error), ME 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 z0.025 = 1.96 ME = 0.03 Estimate P(1 – P) = 0.25 0.25 z n= ME α/2 (0.25)(1.96) = = 1067.11 (0.03) So use n = 1068 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 8-41 8.5 Sample Size Determination: Finite Populations Finite Populations For the Mean Calculate the required sample size n0 using the prior formula: z 2α/2 σ n0 = ME A finite population correction factor is added: σ N−n Var( X) = n N −1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Then adjust for the finite population: n0N n= n0 + (N - 1) Ch 8-42 Sample Size Determination: Finite Populations Finite Populations For the Proportion A finite population correction factor is added: P(1- P) N − n ˆ Var( p) = n N −1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Solve for n: NP(1− P) n= (N − 1)σ p2ˆ + P(1− P) The largest possible value for this expression (if P = 0.25) is: 0.25(1 − P) n= (N − 1)σ p2ˆ + 0.25 A 95% confidence interval will extend ±1.96 σ pˆ from the sample proportion Ch 8-43 Example: Sample Size to Estimate Population Proportion (continued) σ pˆ 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 z0.025 = 1.96 ME = 0.05 1.96 σ pˆ = 0.05 ⇒ σ pˆ = 0.02551 nmax 0.25N (0.25)(850) = = = 264.8 2 (N − 1)σ pˆ + 0.25 (849)(0.02 551) + 0.25 So use n = 265 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 8-45 Chapter Summary Compared two dependent samples (paired samples) Compared two independent samples 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 .. .Chapter Goals After completing this chapter, you should be able to: Form confidence intervals for the difference between two means from dependent samples Form confidence intervals for. .. 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 ... 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):