Statistics for business economics 7th by paul newbold chapter 08

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Statistics for business economics 7th by paul newbold chapter 08

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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):

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Mục lục

  • Slide 1

  • Chapter Goals

  • Estimation: Additional Topics

  • Dependent Samples

  • Mean Difference

  • Confidence Interval for Mean Difference

  • Slide 7

  • Slide 8

  • Slide 9

  • Difference Between Two Means: Independent Samples

  • Slide 11

  • σx2 and σy2 Known

  • Slide 13

  • Confidence Interval, σx2 and σy2 Known

  • σx2 and σy2 Unknown, Assumed Equal

  • Slide 16

  • Slide 17

  • Confidence Interval, σx2 and σy2 Unknown, Equal

  • Pooled Variance Example

  • Calculating the Pooled Variance

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