ECON
14 Sampling Distributions Part 3

# Just rearrange margin of error to get sample size 19

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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

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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

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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 . =

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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 . 0 ˆ = p 801 03 . 0 75 . 0 * 25 . 0 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 . 0 5 . 0 96 . 1 2 2 2 =

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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:
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• Fall '12
• StephenD.Joyce

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