Estimation of Population Parameters Confidence Intervals Learning Objectives

Estimation of Population Parameters: Confidence Intervals

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

Statistical Methods Descriptive Statistics Inferential Statistics Estimation Hypothesis Testing

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

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

Estimation Methods Estimation Point Estimation Confidence Interval Estimation Bootstrapping Prediction Interval

Point Estimation n 1. Provides Single Value n n n 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 Estimation Confidence Interval Estimation Bootstrapping Prediction Interval

Interval Estimation n 1. Provides Range of Values n n 2. Gives Information about Closeness to Unknown Population Parameter n n Based on Observations from 1 Sample Stated in terms of Probability n 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. Confidence Interval Confidence Limit (Lower) Sample Statistic (Point Estimate) 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 sx_ mx-2. 58 s`x mx-1. 65 s`x mx-1. 96 s`x mx mx+1. 65 s`x 90% Samples 95% Samples 99% Samples `X mx+2. 58 s`x mx+1. 96 s`x

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

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

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

Confidence Interval Estimates Confidence Intervals Mean sx Known Proportion sx Unknown Variance Finite Population

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

Estimation Example Mean (s. X Known) mean of a random sample of n = 25 is`X = 50. Set up a 95% confidence interval estimate for m. X if s. X = 10. s. X X - Za / 2 × £ m X £ X + Za / 2 × n n 10 10 50 - 196. ×. × £ m X £ 50 + 196 25 25 46. 08 £ m X £ 53. 92 n. The

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

Confidence Interval Solution* X - Za / 2 × 199. - 1645. × s. X n . 05 100 £ m X £ X + Za / 2 × s. X n . + 1645. £ m X £ 199 × 1982. . £ m X £ 1998 . 05 100

Confidence Interval Estimates Confidence Intervals Mean sx Known Proportion sx Unknown Variance Finite Population

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

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

Student’s t Table Upper Tail Area df . 25 . 10 . 05 a/2 Assume: n=3 df = n - 1 = 2 a =. 10 a/2 =. 05 1 1. 000 3. 078 6. 314 2 0. 817 1. 886 2. 920 . 05 3 0. 765 1. 638 2. 353 t Values 0 2. 920 t

Degrees of Freedom (df) n n 1. Number of Observations that Are Free to Vary After Sample Statistic Has Been Calculated degrees of freedom 2. n 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 = n -1 = 3 -1 =2

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

Thinking Challenge n n 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?

Confidence Interval Solution* n`X = 3. 7 n S = 3. 8987 n n = 6, df = n -1 = 6 -1 = 5 n S / Ön = 3. 8987 / Ö 6 = 1. 592 n t. 05, 5 = 2. 0150 3. 7 - (2. 015)(1. 592) £ m. X £ 3. 7 + (2. 015)(1. 592) n n . 492 £ m. X £ 6. 908

Confidence Interval Estimates Confidence Intervals Mean sx Known Proportion sx Unknown Variance Finite Population

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

Confidence Interval Estimates Confidence Intervals Mean sx Known Proportion sx Unknown Variance Finite Population

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

Estimation Example Proportion n. A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p. ps - Z a / 2 ×. 08 - 196. × ps × (1 - ps ) n. 08 × (1 -. 08 ) 400 £ ps + Z a / 2 ×. × £ p £. 08 + 196 . 053 £ p £. 107 ps × (1 - ps ) n. 08 × (1 -. 08 ) 400

Thinking Challenge n 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?

ü Confidence Interval Solution* n·p ³ 5 n·(1 - p) ³ 5 ps × (1 - ps ) ps - Z a / 2 × £ ps + Z a / 2 × n n. 175 × (. 825). 175 - 1645. . × £ p £. 175 + 1645 × 200. 1308 £ p £. 2192

Estimation Methods Estimation Point Estimation Confidence Interval Estimation Bootstrapping Prediction Interval

Bootstrapping Method n 1. Used If Population Is Not Normal n 2. Requires Computer n 3. Steps n n n Take Initial 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 Estimation Confidence Interval Estimation Bootstrapping Prediction Interval

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

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

Sample Size Example n. 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= 2 Z sx 2 Error 2 (1645 ) ( ). 45 = (5) 2 2 2 = 219. 2 @ 220

Thinking Challenge n 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 s. X was about $400. What sample sizedo you use?

Sample Size Solution* 2 n= Z sx 2 Error 2 (196 ) ( ). 400 = (50) 2 2 2 = 245. 86 @ 246

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

Summary of Interval Estimation Procedures for a Population Mean Yes s known ? Yes No n > 30 ? No Yes Use s to estimate s s known ? Yes No Use s to estimate s Popul. approx. normal ? No Increase n to > 30