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Estimation Confidence Intervals 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 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 Statistics Inferential Statistics Estimation Hypothesis Testing Estimation Process Population Mean, m, is unknown Random Sample Mean `X = 50 I am 95% confident that m is between 40 & 60. Unknown Population Parameters Are Estimated Estimate Population Parameter... Mean m Proportion Variance Differences p s with Sample Statistic `x p 2 m1 - m2 s 2 `x1 -`x2 Estimator and Estimate 1. The Estimator is a Random Variable Used to Estimate a Population Parameter Sample Mean, Sample Proportion, Sample Median Sample Mean X is an Estimator of Population Mean m 2. The estimate is the numerical value of the estimator If X = 3 then 3 Is the Estimate of m Properties of Mean Unbiasedness Efficiency Mean of Sampling Distribution Equals Population Mean Sample Mean Comes Closer to Population Mean Than Any Other Unbiased Estimator Consistency As Sample Size Increases, Variation of Sample Mean from Population Mean Decreases Unbiasedness P(`X) Unbiased Biased A C mx= mx A mx C `X Efficiency P(`X) Sampling Distribution of Mean B Sampling Distribution of Median A mx `X Consistency P(`X) Larger Sample Size B Smaller Sample Size A mx `X Estimation Methods Estimation Point Estimation Confidence Interval Interval Estimation Bootstrapping Prediction Interval Point Estimation Provides Single Value Based on Observations from 1 Sample Gives No Information about How Close Value Is to the Unknown Population Parameter Sample Mean`X = 3 Is Point Estimate of Unknown Population Mean Estimation Methods Estimation Point Estimation Confidence Interval Interval Estimation Bootstrapping Prediction Interval Interval Estimation Provides Range of Values Gives Information about Closeness to Unknown Population Parameter Based on Observations from 1 Sample Stated in terms of Probability – Knowing Exact Closeness Requires Knowing Unknown Population Parameter 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. (1) m X Error (2) Error X - m or X m X -m (3) Z (4) Error Zs x (5) m X Zs x sx Error sx Many Samples Have Same Confidence Interval X = m ± Zs`x sx_ m-2.58s`x m-1.65s`x m-1.96s`x m m+1.65s`x 90% Samples 95% Samples 99% Samples m+2.58s`x m+1.96s`x `X Level of Confidence Probability that the Unknown Population Parameter Falls Within Interval Denoted (1 - a) % a Is Probability That Parameter Is Not Within Interval Typical Values Are 99%, 95%, 90% Intervals & Level of Confidence Sampling Distribution of Mean s_ a/2 1-a a/2 m =m Intervals Extend from `X - Zs`X to `X + Zs`X x x _ X (1 - a) % of Intervals Contain m . a % Do Not. Large Number of Intervals Factors Affecting Interval Width Data Dispersion Measured Sample Size sx = by s Intervals Extend from `X - Zs`X to`X + Zs`X s / n Level of Confidence (1 - a) Affects Z © 1984-1994 T/Maker Co. Confidence Interval Estimates Confidence Intervals Mean s Known Proportion s Unknown Variance Finite Population Confidence Interval Estimate Mean (sX Known) Assumptions Population Standard Deviation Is Known Population Is Normally Distributed If Not Normal, Can Be Approximated by Normal Distribution (n 30) Confidence Interval Estimate s s X - Za / 2 m X Za / 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 s = 10. s s X- Z m X Z n n 50 - 196 . 10 25 . m 50 196 46.08 m 53.92 10 25 Thinking Challenge You’re a Q/C inspector for Gallo. The s for 2liter 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- Z 199 . - 1645 . s n .05 100 m X Z s n . 1645 . m 199 1982 . . m 1998 .05 100 Confidence Interval Estimates Confidence Intervals Mean sx Known Proportion sx Unknown Variance Finite Population Confidence Interval Estimate Mean (s Unknown) Assumptions Population Standard Deviation Is Unknown Population Must Be Normally Distributed n<30 (See comment) Use Student’s t Distribution Confidence Interval Estimate X - t S n m X t 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 Area in Both Tails df .50 .20 Assume: n=3 df = n - 1 = 2 a = .10 a/2 =.05 a .10 1 1.000 3.078 6.314 2 0.817 1.886 2.920 .05 .05 3 0.765 1.638 2.353 t Values 0 2.920 t Degrees of Freedom (df) Number of Observations that Are Free to Vary After Sample Statistic Has Been Calculated degrees of freedom Example Sum of 3 Numbers Is 6 X1 = 1 (or Any Number) X2 = 2 (or Any Number) X3 = 3 (Cannot Vary) Sum = 6 = 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 m. X -t 50 - 2.064 S n 8 25 m X t S n m 50 2.064 46.69 m 53.30 8 25 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? Confidence Interval Solution* X -t S n m X t S n X = (3.6+4.2+4.0+3.5+3.8+3.1)/6 = 3.7 S = .38987 n = 6, df = n -1 = 6 -1 = 5 Sx=S / n = 3.8987 / 6 = .1592 t.05,5 = 2.0150 3.7 - (2.015)(.1592) m 3.7 + (2.015)(.1592) 3.38 m 4.02 Computer Printout De scriptives MINUTES Mean 90% Confidence Int erval for Mean St atis tic 3. 700 Lower Bound Upper Bound St d. Error .159 3. 379 4. 021 5% Trimmed Mean 3. 706 Median Variance St d. Deviation Minimum Maximum Range Int erquartile Range Sk ewness Kurtos is 3. 700 .152 .390 3. 1 4. 2 1. 1 .650 -.364 -.130 .845 1. 741 Confidence Interval Estimates Confidence Intervals Mean sx Known Proportion sx Unknown Variance Finite Population Estimation for Finite Populations Assumptions Sample Is Large Relative to Population – n / N > .05 Use Finite Population Correction Factor Confidence Interval (Mean, s Unknown) X - t S n N-n N -1 m X t S n N-n N -1 Confidence Interval Estimates Confidence Intervals Mean sx Known Proportion sx Unknown Variance Finite Population Confidence Interval Estimate Proportion Assumptions Two Categorical Outcomes Population Follows Binomial Distribution Normal Approximation Can Be Used – n·p 5 & n·(1 - p) 5 Confidence Interval Estimate p -Z p (1 - p ) p p Z n p (1 - p ) 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 - Za / 2 .08 - 196 . p (1 - p ) p p Za / 2 n .08 (1-.08 ) 400 . p .08 196 .053 p .107 p (1 - p ) n .08 (1-.08 ) 400 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* n·p 5 n·(1 - p) 5 p (1 - p ) p (1 - p ) p - Za / 2 p p Za / 2 n n .175 (.825) .175 (.825) .175 - 1645 . . p .175 1645 200 200 .1308 p .2192 Confidence Interval of the Difference between Two Means • Two independent samples • Two large samples - both samples >= 30 • Population standard deviations are unknown • Answer finds the interval: u1 - u2 Confidence Interval of the Difference between Two Means s x1 - x2 n n 1 2 s s 1 2 2 2 where (x - x ) z s 1 2 x1 - x2 Example 6.3, Page 283 Sample 1 Sample 2 x1 = $76 s1 = $25 n1 = 100 x2 = $65 s2 = $22 n2 = 100 s x1 - x2 n1 n2 Sample 1 2 1 s s 2 2 where ( x1 - x ) z s 2 x1 - x2 x1 = $76 s1 = $25 n1 = 100 sx1 )- x 2 Sample 2 x2 = $65 s2 = $22 n2 = 100 (25) 2 (22) 2 3.33 100 100 (76 - 65) 3(3.33) 11 9.99 $1 u1 - u2 $21 Confidence Interval of the Difference between Two Proportions • Two independent samples • Two large samples - both samples >= 30 • Answer finds the interval: p1 - p2 Confidence Interval of the Difference between Two Proportions p - p ) z s 2 1 p1 - p 2 where sp -p 1 2 p1 q1 p 2 q 2 n2 n1 Selecting a Sample Size Selecting a Sample Size The Degree of Confidence Selected The Maximum Allowable Error The Population Standard Deviation Finding Sample Sizes (1) (2) (3) Z X - mx sx Error sx Error Zs x Z n 2 Z sx 2 Error 2 sx n I don’t want to sample too much or too little! Sample Size for Means z s z s n 2 E E 2 2 2 E is the allowable error z is the z score associated with degree of confidence s is the population standard deviation The marketing manager would like to estimate the population mean annual usage of home heating oil to within 50 gallons of the true value and desires to be 95% confident of correctly estimating the true mean. Based on a previous study taken last year,the marketing manager feels that the standard deviation can be estimated as 325 gallons. What is the sample size need to obtain these results? z = 1.96 Confidence = 95% E = 50 s = 325 196 z s . ) (325) (384 . )(105,625) n 162.31 2 2 2500 E (50) 2 2 2 2 n 163 homes need to be sampled Sample Size for Proportions n p 1 - p) z E 2 2 E is the maximum allowable error z is the z value associated with the degree of confidence p is the estimated proportion A political pollister would like to estimate the proportion of voters who will vote for the Democratic candidate in a presidential campaign. The pollster would like 95% confidence that her prediction is correct to within .04 of the true proportion. What sample size is needed? Z=1.96 Confidence = 95% E = .04 p = unknown use p = .5 p(1 - P) z .5(1-.5)(196 . ) n 600.25 2 2 E (.04) 2 n = 601 voters 2 Conclusion Stated What Is Estimated Distinguished Point & Interval Estimates Explained Interval Estimates Computed Confidence Interval Estimates for Population Mean & Proportion Computed Sample Size