14 Sampling Distributions Part 3

P 176621 0283 ma rg in o f e rro r 196 2831 283621 9

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p = 176/621 = ̂ 0.283 Ma rg in o f e rro r = 1.96 *√(.283*(1-.283)/621) 9 5 % CI: (.2 4 8 , . 3 8 1 ) p ̂ + m a rg in o f e rro r = .2 8 3 + 0 .0 3 5 = 0 .0 3 5 12

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Summary: Computing (1-α)*100% confidence intervals Proportion = sample proportion np>5; n(1-p)>5 Mean, σ unknown s = sample standard deviation t has n-1 degrees of freedom n ≥ 30 if population not normally distributed Mean, σ known n ≥ 30 if population is not normally distributed 13 n p p z p ) ˆ 1 ( ˆ ˆ ) 2 / ( - ± α n s t x ) 2 / ( α ± n z x σ α ) 2 / ( ± ˆ p
Determining sample size Before collecting data, need to determine the sample size n for estimating the population mean or proportion. One method is to collect enough data so that the 95% confidence interval will have a predetermined margin of error 14

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Margin of Error (e): Our tolerance level for sampling error. Margin of error mples: σ = 25, n=300, and 95% conf. level: e = σ = 25, n=50, and 95% conf. level: e = 1.96 * 25 / 300 = 2.829 6.930 Determining sample size: Margin of error 15 n σ z e α/ ) 2 ( = n /2) ( σ α z x ±
Margin of Error (e): Confidence Interval 1. Data variation, σ or s : e as σ 2. Sample size, n : e as n 3. Level of confidence, 1 - : e if 1 - or or or Which is bigger? 1. s = 10,  = 0.05, e when n =50 vs e when n =500 2. n = 50,  = 0.05, e when s =50 vs e when s =75 3. n = 50, s = 250 e when =0.01vs e when =0.05 Determining sample size: Margin of error 16 n σ z x ) 2 / ± n σ z e ) 2 / =

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Just rearrange margin of error to get sample size Required sample size, σ known: What if σ is unknown? Estimate σ - Select a pilot sample and estimate σ with the sample standard deviation s - If range, R, is known, may use σ = R/6 (Recall Empirical Rule, ±3σ contains virtually all data) Example Always round-up! Determining sample size, n 17 2 ) 2 / ( = e z n σ α 5 100 96 . 1 ) 2 ( = = = e z σ α = n 1537 5 100 * 96 . 1 2 =
Example: Communications expenditures Want an estimate of mean monthly expenditure on communications Desire a margin of error of \$5 for the mean, at a 95% confidence level. From other work, estimate S = \$30 What should you use as a sample size? n = (1.96 * 30/ 5)2 = 138.3  use 139 18 2 ) 2 / ( = e z n σ α

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Margin of Error (e): Confidence Interval Determining sample size for proportions : Margin of error Finding sample size: Do the same thing for finding required sample size for proportions.
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