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EF 507 QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE FALL 2008 Chapter 9

EF 507 QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE FALL 2008 Chapter 9 Estimation: Additional Topics. Estimation: Additional Topics. Chapter Topics. Population Means, Dependent Samples. Population Means, Independent Samples. Population Proportions. Population Variance.

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EF 507 QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE FALL 2008 Chapter 9

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  1. EF 507 QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE FALL 2008 Chapter 9 Estimation: Additional Topics

  2. 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

  3. 1. 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 • di = xi - yi

  4. Mean Difference The ith paired difference is di , where 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

  5. Confidence Interval forMean 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)

  6. Confidence Interval forMean 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

  7. Paired Samples Example • Six people sign up for a weight loss program. You collect the following data:  Weight: PersonBefore (x)After (y)Difference,di 1 136125 11 2 205195 10 3 157150 7 4 138140 - 2 5 175165 10 6 166160 6 42 di d = n = 7.0

  8. Paired Samples Example (continued) • For a 95% confidence level, the appropriate t value is tn-1,/2 = t5,0.025 = 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

  9. 2. Difference Between Two Means Goal: Form a confidence interval for the difference between two population means, μx – μy • 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: Population means, independent samples x – y

  10. Difference Between Two Means (continued) Population means, independent samples σx2 and σy2known Confidence interval uses z/2 σx2 and σy2unknown σx2 and σy2 assumed equal Confidence interval uses a value from the Student’s t distribution σx2 and σy2 assumed unequal

  11. 2. σx2 and σy2Known 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

  12. σx2 and σy2Known (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

  13. Confidence Interval, σx2 and σy2Known Population means, independent samples * σx2 and σy2 known The confidence interval for μx – μy is: σx2 and σy2 unknown

  14. 3.a) σx2 and σy2Unknown, 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

  15. σx2 and σy2Unknown,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

  16. σx2 and σy2Unknown,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

  17. Confidence Interval, σx2 and σy2Unknown, Equal σx2 and σy2 unknown * σx2 and σy2 assumed equal The confidence interval for μ1 – μ2 is: σx2 and σy2 assumed unequal Where

  18. 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): CPUxCPUyNumber Tested 17 14 Sample mean 3,004 2,538 Sample std dev 74 56 Pooled Variance Example Assume both populations are normal with equal variances, and use 95% confidence

  19. Calculating the Pooled Variance The pooled variance is: The t value for a 95% confidence interval is:

  20. 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.

  21. 3.b) σx2 and σy2Unknown,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

  22. σx2 and σy2Unknown,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

  23. Confidence Interval, σx2 and σy2Unknown, Unequal σx2 and σy2 unknown σx2 and σy2 assumed equal The confidence interval for μ1 – μ2 is: * σx2 and σy2 assumed unequal Where

  24. 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

  25. Two Population Proportions (continued) • The random variable is approximately normally distributed Population proportions

  26. Confidence Interval forTwo Population Proportions Population proportions The confidence limits for Px – Py are:

  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. • In a random sample, 26 of 50 men and 28 of 40 women had an earned college degree

  28. Example: Two Population Proportions (continued) Men: Women: For 90% confidence, Z/2 = 1.645

  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

  30. Confidence Intervals for the Population Variance • The confidence interval is based on the sample variance, s2 • Assumed: the population is normally distributed Population Variance • Goal: Form a confidence interval for the population variance,σ2

  31. 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

  32. Confidence Intervals for the Population Variance (continued) Population Variance The (1 - )% confidence interval for the population variance is

  33. You are testing the speed of a computer processor. You collect the following data (in Mhz): CPUxSample size 17 Sample mean 3,004 Sample std dev 74 Example Assume the population is normal. Determine the 95% confidence interval for σx2

  34. 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 =0 .025 probability α/2 = 0.025 216 216 216 = 6.91 = 28.85

  35. 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

  36. Sample PHStat Output

  37. Sample PHStat Output (continued) Input Output

  38. Sample Size Determination Determining Sample Size For the Mean For the Proportion

  39. 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

  40. Sample Size Determination Determining Sample Size For the Mean Margin of Error (sampling error)

  41. Sample Size Determination (continued) Determining Sample Size For the Mean Now solve for n to get

  42. 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

  43. 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)

  44. Sample Size Determination Determining Sample Size For the Proportion Margin of Error (sampling error)

  45. Sample Size Determination (continued) Determining Sample Size For the Proportion Substitute 0.25 for and solve for n to get cannot be larger than 0.25, when = 0.5

  46. 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

  47. 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?

  48. 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 = 1,068

  49. PHStat Sample Size Options

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