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Statistics for Business and Economics Estimation of Population Parameters: Confidence Intervals Chapter 8

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Learning Objectives 1.State What Is Estimated 1.State What Is Estimated 2.Distinguish Point & Interval Estimates 2.Distinguish Point & Interval Estimates 3.Explain Interval Estimates 3.Explain Interval Estimates 4.Compute Confidence Interval Estimates for Population Mean & Proportion 4.Compute Confidence Interval Estimates for Population Mean & Proportion 5.Compute Sample Size 5.Compute Sample Size

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Thinking Challenge Suppose you’re interested in the average amount of money that students in this class (the population) have on them. How would you find out? Suppose you’re interested in the average amount of money that students in this class (the population) have on them. How would you find out?

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Statistical Methods Statistical Methods Descriptive Statistics Inferential Statistics Estimation Hypothesis Testing

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Estimation Process Mean, X, is unknown PopulationRandom Sample I am 95% confident that X is between 40 & 60. Mean X = 50

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Unknown Population Parameters Are Estimated Estimate Population Parameter... with Sample Statistic Mean x x Proportionp p s Variance x 2 s 2 Differences 1 2 x 1 - x 2

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Estimation Methods Estimation Point Estimation Interval Estimation Confidence Interval Prediction Interval Boot- strapping

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Point Estimation 1.Provides Single Value 1.Provides Single Value Based on Observations from 1 Sample Based on Observations from 1 Sample 2.Gives No Information about How Close Value Is to the Unknown Population Parameter 2.Gives No Information about How Close Value Is to the Unknown Population Parameter 3.Sample Mean X = 3 Is Point Estimate of Unknown Population Mean 3.Sample Mean X = 3 Is Point Estimate of Unknown Population Mean

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Estimation Methods Estimation Point Estimation Interval Estimation Confidence Interval Prediction Interval Boot- strapping

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Interval Estimation 1.Provides Range of Values 1.Provides Range of Values Based on Observations from 1 Sample Based on Observations from 1 Sample 2.Gives Information about Closeness to Unknown Population Parameter 2.Gives Information about Closeness to Unknown Population Parameter Stated in terms of Probability Stated in terms of Probability Knowing Exact Closeness Requires Knowing Unknown Population Parameter Knowing Exact Closeness Requires Knowing Unknown Population Parameter 3.e.g., Unknown Population Mean Lies Between 50 & 70 with 95% Confidence 3.e.g., Unknown Population Mean Lies Between 50 & 70 with 95% Confidence

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Key Elements of Interval Estimation Confidence Interval Sample Statistic (Point Estimate) Confidence Limit (Lower) Confidence Limit (Upper) A Probability That the Population Parameter Falls Somewhere Within the Interval.

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Confidence Limits for Population Mean Parameter = Statistic ± Error © 1984-1994 T/Maker Co.

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Many Samples Have Same Confidence Interval 90% Samples 95% Samples 99% Samples x +1.65 x x +2.58 x x _ XX x +1.96 x x -2.58 x x -1.65 x x -1.96 x xx X = x ± Z x

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1.Probability that the Unknown Population Parameter Falls Within Interval 1.Probability that the Unknown Population Parameter Falls Within Interval 2.Denoted (1 - 2.Denoted (1 - Is Probability That Parameter Is Not Within Interval Is Probability That Parameter Is Not Within Interval 3.Typical Values Are 99%, 95%, 90% 3.Typical Values Are 99%, 95%, 90% Level of Confidence

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Intervals & Level of Confidence Sampling Distribution of Mean Large Number of Intervals Intervals Extend from X - Z X to X + Z X (1 - ) % of Intervals Contain X. % Do Not. x = x 1 - /2 X _ x _

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Factors Affecting Interval Width 1.Data Dispersion 1.Data Dispersion Measured by X Measured by X 2.Sample Size 2.Sample Size X = X / n X = X / n 3.Level of Confidence (1 - ) 3.Level of Confidence (1 - ) Affects Z Affects Z Intervals Extend from X - Z X to X + Z X © 1984-1994 T/Maker Co.

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Confidence Interval Estimates ProportionMean x Unknown Confidence Intervals Variance Finite Population x Known

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Confidence Interval Estimate Mean ( X Known) 1.Assumptions 1.Assumptions Population Standard Deviation Is Known Population Standard Deviation Is Known Population Is Normally Distributed Population Is Normally Distributed If Not Normal, Can Be Approximated by Normal Distribution (n 30) If Not Normal, Can Be Approximated by Normal Distribution (n 30) 2.Confidence Interval Estimate 2.Confidence Interval Estimate XZ n XZ n X X X //22

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Estimation Example Mean ( X Known) The mean of a random sample of n = 25 is X = 50. Set up a 95% confidence interval estimate for X if X = 10. The mean of a random sample of n = 25 is X = 50. Set up a 95% confidence interval estimate for X if X = 10. XZ n XZ n X X X X X //.... 22 50196 10 25 50196 10 25 46085392

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Thinking Challenge You’re a Q/C inspector for Gallo. The X for 2- liter bottles is.05 liters. A random sample of 100 bottles showed X = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles? You’re a Q/C inspector for Gallo. The X for 2- liter bottles is.05 liters. A random sample of 100 bottles showed X = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles? 2 liter © 1984-1994 T/Maker Co.

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Confidence Interval Solution* XZ n XZ n X X X X X //........ 22 1991645 05 100 1991645 05 100 19821998

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Confidence Interval Estimates ProportionMean x Unknown Confidence Intervals Variance Finite Population x Known

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Confidence Interval Estimate Mean ( X Unknown) 1.Assumptions 1.Assumptions Population Standard Deviation Is Unknown Population Standard Deviation Is Unknown Population Must Be Normally Distributed Population Must Be Normally Distributed 2.Use Student’s t Distribution 2.Use Student’s t Distribution 3.Confidence Interval Estimate 3.Confidence Interval Estimate Xt S n Xt S n nXn /,/,2121

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Z t Student’s t Distribution 0 t (df = 5) Standard Normal t (df = 13) Bell-Shaped Symmetric ‘Fatter’ Tails

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Upper Tail Area df.25.10.05 11.0003.0786.314 2 0.8171.886 2.920 30.7651.6382.353 t 0 Student’s t Table Assume: n = 3 df= n - 1 = 2 =.10 /2 =.05 2.920 t Values / 2.05

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Degrees of Freedom (df) 1.Number of Observations that Are Free to Vary After Sample Statistic Has Been Calculated 1.Number of Observations that Are Free to Vary After Sample Statistic Has Been Calculated 2.Example 2.Example Sum of 3 Numbers Is 6 X 1 = 1 (or Any Number) X 2 = 2 (or Any Number) X 3 = 3 (Cannot Vary) Sum = 6 Sum of 3 Numbers Is 6 X 1 = 1 (or Any Number) X 2 = 2 (or Any Number) X 3 = 3 (Cannot Vary) Sum = 6 degrees of freedom = n -1 = 3 -1 = 2

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Estimation Example Mean ( X Unknown) A random sample of n = 25 has X = 50 & S = 8. Set up a 95% confidence interval estimate for X. A random sample of n = 25 has X = 50 & S = 8. Set up a 95% confidence interval estimate for X. Xt S n Xt S n nXn X X /,/,.... 2121 5020639 8 25 5020639 8 25 46695330

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Thinking Challenge You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time? What is the 90% confidence interval estimate of the population mean task time?

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Confidence Interval Solution* X = 3.7 X = 3.7 S = 3.8987 S = 3.8987 n = 6, df = n -1 = 6 -1 = 5 n = 6, df = n -1 = 6 -1 = 5 S / n = 3.8987 / 6 = 1.592 S / n = 3.8987 / 6 = 1.592 t.05,5 = 2.0150 t.05,5 = 2.0150 3.7 - (2.015)(1.592) X 3.7 + (2.015)(1.592) 3.7 - (2.015)(1.592) X 3.7 + (2.015)(1.592).492 X 6.908.492 X 6.908

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Confidence Interval Estimates ProportionMean x Unknown Confidence Intervals Variance Finite Population x Known

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Estimation for Finite Populations 1.Assumptions 1.Assumptions Sample Is Large Relative to Population Sample Is Large Relative to Population n / N >.05 n / N >.05 2.Use Finite Population Correction Factor 2.Use Finite Population Correction Factor 3.Confidence Interval (Mean, X Unknown) 3.Confidence Interval (Mean, X Unknown) Xt S n Nn N Xt S n Nn N nXn /,/,2121 11

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Confidence Interval Estimates ProportionMean x Unknown Confidence Intervals Variance Finite Population x Known

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Confidence Interval Estimate Proportion 1.Assumptions 1.Assumptions Two Categorical Outcomes Two Categorical Outcomes Population Follows Binomial Distribution Population Follows Binomial Distribution Normal Approximation Can Be Used Normal Approximation Can Be Used n·p 5 & n·(1 - p) 5 n·p 5 & n·(1 - p) 5 2.Confidence Interval Estimate 2.Confidence Interval Estimate pZ pp n ppZ pp n s ss s ss ()()11

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Estimation Example Proportion A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p. A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p. pZ pp n ppZ pp n p p s ss s ss // ()()...(.)...(.).. 22 11 08196 081 400 08196 081 400 053107

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Thinking Challenge You’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective? You’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective?

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Confidence Interval Solution* n·p 5 n·(1 - p) 5 pZ pp n ppZ pp n p p s ss s ss // ()()...(.)... ).. 22 11 1751645 175825 200 1751645 175825 200 13082192

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Estimation Methods Estimation Point Estimation Interval Estimation Confidence Interval Prediction Interval Boot- strapping

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Bootstrapping Method 1.Used If Population Is Not Normal 1.Used If Population Is Not Normal 2.Requires Computer 2.Requires Computer 3.Steps 3.Steps Take Initial Sample Take Initial Sample Sample Repeatedly from Initial Sample Sample Repeatedly from Initial Sample Compute Sample Statistic Compute Sample Statistic Form Resampling Distribution Form Resampling Distribution Limits Are Values That Cut Off Smallest & Largest /2 % Limits Are Values That Cut Off Smallest & Largest /2 %

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Estimation Methods Estimation Point Estimation Interval Estimation Confidence Interval Prediction Interval Boot- strapping

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Prediction Interval 1.Used to Estimate Future Individual X Value 1.Used to Estimate Future Individual X Value 2.Not Used to Estimate Unknown Population Parameter 2.Not Used to Estimate Unknown Population Parameter 3.Prediction Interval Estimate 3.Prediction Interval Estimate

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Finding Sample Sizes I don’t want to sample too much or too little! (1) (2) Z XError ZZ n n Z x xx x x x ()3 2 2 2

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Sample Size Example What sample size is needed to be 90% confident of being correct within 5? A pilot study suggested that the standard deviation is 45. What sample size is needed to be 90% confident of being correct within 5? A pilot study suggested that the standard deviation is 45. n Z Error x 2 2 2 22 2 164545 5 2192220 .. ()() ()

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Thinking Challenge You work in Human Resources at Merrill Lynch. You plan to survey employees to find their average medical expenses. You want to be 95% confident that the sample mean is within ± $50. A pilot study showed that X was about $400. What sample size do you use? You work in Human Resources at Merrill Lynch. You plan to survey employees to find their average medical expenses. You want to be 95% confident that the sample mean is within ± $50. A pilot study showed that X was about $400. What sample size do you use?

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Sample Size Solution* n Z Error x 2 2 2 22 2 196400 50 24586246 .. ()() ( )

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Conclusion 1.Stated What Is Estimated 1.Stated What Is Estimated 2.Distinguished Point & Interval Estimates 2.Distinguished Point & Interval Estimates 3.Explained Interval Estimates 3.Explained Interval Estimates 4.Computed Confidence Interval Estimates for Population Mean & Proportion 4.Computed Confidence Interval Estimates for Population Mean & Proportion 5.Computed Sample Size 5.Computed Sample Size

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