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14 Sampling Distributions Part 3

# Just rearrange margin of error to get sample size 19

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Unformatted text preview: Just rearrange margin of error to get sample size. 19 n p p z e ) 1 ( /2) (- = α n p p z p ) 1 ( ) 2 / (- ± α 2 2 /2) ( ) 1 ( e p p z n- = α Don’t know proportion: Make a guess p is unknown Can be estimated with a pilot sample Use prior knowledge You may know that 2% to 7% of all accounts receivables have errors. Use p = 0.07 (value closer to ½) You may know that service availability is between 98% to 99.9%. Use p = .98. Are you clueless? Use p = .50? Gives too many observations 20 Example: Election polling Polling for presidential election between 2 candidates 95% confidence interval Margin of error = + 2% Guess at proportion = 0.5 n = n = [1.962 *.5(.5)]/(.02)2 = 2,401 21 2 2 /2) ( ) 1 ( e p p z n- = α Example: Election polling Polling for presidential election between 2 candidates 95% confidence interval Margin of error = + 4% Guess at proportion = 0.5 n = [1.962 *.5(.5)]/(.04)2 = 601 22 2 2 /2) ( ) 1 ( e p p z n- = α The Business School is designing a new auditorium and is interested in knowing how many people are left handed so that they can allocate enough seating space for them. A random sample of 100 people shows that 25 are left-handed. 1. Form a 95% confidence interval for the true proportion of left- handers 0.25 = 0.25 ± 1.96 * 0.0433 = [0.1651, 0.3349] Example: Left-handers 23 =- = = n p p p ) ˆ 1 ( ˆ error std. ˆ p ˆ σ n p p z p ) ˆ 1 ( ˆ ˆ /2) (- ± α 0433 . 100 75 . * 25 . = 2. How large a sample would be necessary to estimate the true proportion of left-handers within 3%, with 95% confidence? Recall the result of a pilot study Margin of error (e) = 0.03 Example: Left-handers 24 =- = 2 2 /2 ) 1 ( e p p z n α 25 . ˆ = p 801 03 . 75 . * 25 . 96 . 1 2 2 = 3. There were some questions raised about the pilot study. How large must the sample be if we want the margin of error for a 95% confidence interval to be less than 0.01, regardless of the true value of p? Margin of error (e) = .01 What value of p give you the most conservative estimate? p=0.5 Example: Left-handers 25 =- = 2 2 /2 ) 1 ( e p p z n α 9604 01 . 5 . 96 . 1 2 2 2 = Important Note – Finite Population Correction Factor Inference presented in these slides assume that the number in the population is very large (infinite). If your sample is larger than 5% of the population, use finite sample correction. 26 where N = population size, n = sample size σ x = σ X n N- n N- 1 σ ˆ p = p (1- p ) n N- n N- 1 Key learning points: Confidence intervals General form of a confidence interval: Point estimate + margin of error Margin of error depends on: Confidence level, (1-α)*100%, which determines z or t Standard deviation of population Sample size, n Means, σ known or unknown Proportions Finding required sample size for a specified margin of error. 27...
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