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Chapter Topics •Confidence Interval Estimation for the Mean (s Known) •Confidence Interval Estimation for the Mean (s Unknown) •Confidence Interval Estimation for the Proportion •Sample Size Estimation Estimation Process Population Mean, m, is unknown Sample Random Sample Mean X = 50 I am 95% confident that m is between 40 & 60. Population Parameters Estimated Estimate Population Parameter... Mean m Proportion Variance Difference with Sample Statistic _ X p s 2 m - m 1 ps s 2 2 _ _ x - x 1 2 Confidence Interval Estimation • Provides Range of Values – Based on Observations from 1 Sample • Gives Information about Closeness to Unknown Population Parameter • Stated in terms of Probability Never 100% Sure Elements of Confidence Interval Estimation A Probability That the Population Parameter Falls Somewhere Within the Interval. Sample Confidence Interval Statistic Confidence Limit (Lower) Confidence Limit (Upper) Level of Confidence • • • Probability that the unknown population parameter falls within the interval • Denoted (1 - a) % = level of confidence e.g. 90%, 95%, 99% a Is Probability That the Parameter Is Not Within the Interval Intervals & Level of Confidence Sampling Distribution of the Mean a/2 Intervals Extend from s_ x 1-a mX m a/2 _ X (1 - a) % of Intervals Contain m. X ZsX to a % Do Not. X ZsX Confidence Intervals Factors Affecting Interval Width • • Data Variation measured by s • Sample Size sX sX / n • Level of Confidence Intervals Extend from X - Zs x to X + Z s x (1 - a) © 1984-1994 T/Maker Co. Confidence Interval Estimates Confidence Intervals Mean s Known Proportion s Unknown Confidence Intervals (s Known) • • Assumptions – Population Standard Deviation Is Known – Population Is Normally Distributed – If Not Normal, use large samples Confidence Interval Estimate s m X Za / 2 n s X Za / 2 n Confidence Intervals (s Unknown) • Assumptions – – Population Standard Deviation Is Unknown Population Must Be Normally Distributed • Use Student’s t Distribution • Confidence Interval Estimate S S m X t X ta / 2 ,n1 a / 2 ,n1 n n Student’s t Distribution Standard Normal Bell-Shaped Symmetric ‘Fatter’ Tails t (df = 13) t (df = 5) 0 Z t Degrees of Freedom (df) • • • Number of Observations that Are Free to Vary After Sample Mean Has Been Calculated • Example – Mean of 3 Numbers Is 2 X1 = 1 (or Any Number) X2 = 2 (or Any Number) X3 = 3 (Cannot Vary) Mean = 2 degrees of freedom = n -1 = 3 -1 =2 Student’s t Table a/2 Upper Tail Area df .25 .10 .05 Assume: n = 3 =n-1=2 df 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 0 t Values 2.920 t Example: Interval Estimation s Unknown •A random sample of n = 25 hasX = 50 and •s = 8. Set up a 95% confidence interval estimate for m. S S X ta / 2 ,n1 m X ta / 2 ,n1 n n 50 2 . 0639 8 25 m 46 . 69 m 50 2 . 0639 53 . 30 8 25 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 ps ( 1 ps ) ps Za / 2 n p ps ( 1 ps ) ps Za / 2 n Sample Size Too Big: •Requires too much resources Too Small: •Won’t do the job Example: Sample Size for Mean •What sample size is needed to be 90% confident of being correct within ± 5? A pilot study suggested that the standard deviation is 45. Z s 2 n 2 Error 2 1645 . 5 2 2 45 2 219.2 @ 220 Round Up