A survey of 100 Oregonians reports that 48% are excited by the opportunity to ta

Question

A survey of 100 Oregonians reports that 48% are excited by the opportunity to take a statistics class Co struct a 95% confidence interval on the true pro ort who are excited to take a statistics class a of Oregon as Since your interval contains values above 50% it seems plausible that more than half ofthe state feels this way, and you would like to explore that idea a bit urther Suppose you decide that you want to refine your estimate of the population proportion and cut the width of your interval in half. Will doubling your sample size do this? How large a sample will be n ded to cut your inte al dth in half? Howlarge a s mple will be needed to shrink your interval to the point where50% will not be included in a 95% confidence interval centered at the 0.48 point estimate?

Explanation / Answer

Solution:

We first check that the sample size is large enough to apply the normal approximation. The true value of p is unknown, so we can't check that np > 10 and n(1-p) > 10, but we can check this for p-hat, our estimate of p. 1000*0.48 = 480 > 10 and 1000*.52 > 10. This means the normal approximation will be good, and we can apply them to calculate a confidence interval for p.

0.48 +/- 1.96*sqrt(.48*.52/1000)

0.48 +/- 0.03096552 (that mysterious 3% margin of error!)

(0.45, 0.51) is a 95% CI for the true proportion of all Californians who are excited about the Stats teachers' visit.
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The current interval width is about 6%. So the current margin of error is 3%. We want margin of error = 1.5% or

1.96*sqrt(.48*.52/n) = .015

Solve for n: n = (1.96/.015)^2 * .48*.52 = 4261.6

We'd need at least 4262 people in the sample. So to cut the width of the CI in half, we'd need about four times as many people.

Assuming that the true value of p = .48, how many people would we need to make sure our CI doesn't include .50? This means the margin of error must be less than 2%, so solving for n:

n = (1.96/.02)^2 *.48*.52 = 2397.1

We'd need about 2398 people.

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