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

Learning Objectives 1. State What Is Estimated 2. Distinguish Point & Interval Estimates 3. Explain Interval Estimates 4. Compute Confidence Interval Estimates for Population Mean & Proportion 5. Compute Sample Size As a result of this class, you will be able to ...

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?

Statistical Methods Statistical Methods Descriptive Inferential Statistics Statistics Hypothesis Estimation Testing

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

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

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

Point Estimation 1. Provides Single Value Based on Observations from 1 Sample 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

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

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

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

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

Many Samples Have Same Confidence Interval `X = mx ± Zs`x s x _ `X mx-2.58s`x mx-1.65s`x mx mx+1.65s`x mx+2.58s`x mx-1.96s`x mx+1.96s`x 90% Samples 95% Samples 99% Samples

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

Intervals & Level of Confidence _ Sampling Distribution of Mean s x a /2 a /2 1 - a _ Notice that the interval width is determined by 1-a in the sampling distribution. X m = m ` x x Intervals Extend from `X - Zs`X to `X + Zs`X (1 - a) % of Intervals Contain mX . a % Do Not. Large Number of Intervals

Factors Affecting Interval Width 1. Data Dispersion Measured by sX 2. Sample Size s`X = sX / Ön 3. Level of Confidence (1 - a) Affects Z Intervals Extend from `X - Zs`X to`X + Zs`X Have students explain why each of these occurs. Level of confidence can be seen in the sampling distribution. © 1984-1994 T/Maker Co.

Confidence Interval Estimates Intervals Mean Proportion Variance Finite s Known s Unknown x x Population

Confidence Interval Estimate Mean (sX Known) 1. Assumptions Population Standard Deviation Is Known Population Is Normally Distributed If Not Normal, Can Be Approximated by Normal Distribution (n ³ 30) 2. Confidence Interval Estimate s s X X X - Z × £ m £ X + Z × a / 2 X a / 2 n n

Estimation Example Mean (sX Known) The mean of a random sample of n = 25 is`X = 50. Set up a 95% confidence interval estimate for mX if sX = 10. s s X X X - Z × £ m £ X + Z × a / 2 X a / 2 n n 10 10 50 - 1 . 96 × £ m £ 50 + 1 . 96 × X 25 25 46 . 08 £ m £ 53 . 92 X

Thinking Challenge You’re a Q/C inspector for Gallo. The sX 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.

Confidence Interval Solution* X X X - Z × £ m £ X + Z × a / 2 X a / 2 n n . 05 . 05 1 . 99 - 1 . 645 × £ m £ 1 . 99 + 1 . 645 × X 100 100 1 . 982 £ m £ 1 . 998 X

Confidence Interval Estimates Intervals Mean Proportion Variance Finite s Known s Unknown x x Population

Confidence Interval Estimate Mean (sX Unknown) 1. Assumptions Population Standard Deviation Is Unknown Population Must Be Normally Distributed 2. Use Student’s t Distribution 3. Confidence Interval Estimate S S X - t × £ m £ X + t × a / 2 , n - 1 X a / 2 , n - 1 n n

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

Student’s t Table .05 2 t a / 2 .05 2.920 t Values Upper Tail Area df Assume: n = 3 df = n - 1 = 2 a = .10 a/2 =.05 a / 2 Upper Tail Area df .25 .10 .05 Confidence intervals use a/2, so divide a! 1 1.000 3.078 6.314 2 0.817 1.886 2.920 .05 3 0.765 1.638 2.353 t 2.920 t Values

Degrees of Freedom (df) 1. Number of Observations that Are Free to Vary After Sample Statistic Has Been Calculated 2. Example Sum of 3 Numbers Is 6 X1 = 1 (or Any Number) X2 = 2 (or Any Number) X3 = 3 (Cannot Vary) Sum = 6 degrees of freedom = n -1 = 3 -1 = 2

Estimation Example Mean (sX Unknown) A random sample of n = 25 has`X = 50 & S = 8. Set up a 95% confidence interval estimate for mX. S S X - t × £ m £ X + t × a / 2 , n - 1 X a / 2 , n - 1 n n 8 8 50 - 2 . 0639 × £ m £ 50 + 2 . 0639 × X 25 25 46 . 69 £ m £ 53 . 30 X

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. What is the 90% confidence interval estimate of the population mean task time? Allow students about 20 minutes to solve.

Confidence Interval Solution* `X = 3.7 S = 3.8987 n = 6, df = n -1 = 6 -1 = 5 S / Ön = 3.8987 / Ö6 = 1.592 t.05,5 = 2.0150 3.7 - (2.015)(1.592) £ mX £ 3.7 + (2.015)(1.592) .492 £ mX £ 6.908

Confidence Interval Estimates Intervals Mean Proportion Variance Finite s Known s Unknown x x Population

Estimation for Finite Populations 1. Assumptions Sample Is Large Relative to Population n / N > .05 2. Use Finite Population Correction Factor 3. Confidence Interval (Mean, sX Unknown) S N - n S N - n X - t × × £ m £ X + t × × a / 2 , n - 1 X a / 2 , n - 1 n N - 1 n N - 1

Confidence Interval Estimates Intervals Mean Proportion Variance Finite s Known s Unknown x x Population

Confidence Interval Estimate Proportion 1. Assumptions Two Categorical Outcomes Population Follows Binomial Distribution Normal Approximation Can Be Used n·p ³ 5 & n·(1 - p) ³ 5 2. Confidence Interval Estimate p × ( 1 - p ) p × ( 1 - p ) s s s s p - Z × £ p £ p + Z × s s n n

Estimation Example Proportion A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p. p × ( 1 - p ) p × ( 1 - p ) s s s s p - Z × £ p £ p + Z × s a / 2 s a / 2 n n . 08 × ( 1 - . 08 ) . 08 × ( 1 - . 08 ) . 08 - 1 . 96 × £ p £ . 08 + 1 . 96 × 400 400 . 053 £ p £ . 107

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?

Confidence Interval Solution* ü n·p ³ 5 n·(1 - p) ³ 5 p × ( 1 - p ) p × ( 1 - p ) s s s s p - Z × £ p £ p + Z × s a / 2 s a / 2 n n . 175 × (. 825 ) . 175 × (. 825 ) . 175 - 1 . 645 × £ p £ . 175 + 1 . 645 × 200 200 . 1308 £ p £ . 2192

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

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

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

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

I don’t want to sample too much or too little! Finding Sample Sizes I don’t want to sample too much or too little! X - m Error x (1) Z = = s s x x s x (2) Error = Z s = Z x n 2 2 Z s x ( 3 ) n = 2 Error

( ) ( ) ( ) 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. ( ) ( ) 2 2 2 2 Z s 1 . 645 45 x n = = ( ) = 219 . 2 @ 220 2 2 Error 5

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 sX was about $400. What sample size do you use?

( ) ( ) ( ) Sample Size Solution* Z s n = Error 1 . 96 400 = 50 = 245 x n = 2 Error ( ) ( ) 2 2 1 . 96 400 = ( ) 2 50 = 245 . 86 @ 246

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

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