SOLUTION: A researcher wishes to estimate the proportion of college students who own a car. He wants to be 95% confident and be accurate within 8% of the true proportion. find the minimum sa
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Question 1192241: A researcher wishes to estimate the proportion of college students who own a car. He wants to be 95% confident and be accurate within 8% of the true proportion. find the minimum sample necessary. Answer by math_tutor2020(3817) (Show Source):
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In your stats class, you should have come across the margin of error for a confidence interval involving proportions.
That particular margin of error formula is:
E = z*sqrt(phat*(1-phat)/n)
Let's solve for n
E = z*sqrt(phat*(1-phat)/n)
sqrt(phat*(1-phat)/n) = E/z
phat*(1-phat)/n = (E/z)^2
n/( phat*(1-phat) ) = (z/E)^2
n = phat*(1-phat)(z/E)^2
At 95% confidence, the z critical value is about z = 1.96 which is determined through a Z table or a calculator.
Unfortunately we don't know what phat is, so we'll have to go with the conservative estimate of phat = 0.5
The desired error we want is E = 0.08, since we want to be within 8% of the true proportion p.
The phrasing "within" is the same as saying "at most". So we want the error to be at most 0.08, ie we want such that E is positive.
Plug in those items mentioned
n = phat*(1-phat)(z/E)^2
n = 0.5*(1-0.5)(1.96/0.08)^2
n = 150.0625
Despite this value of n being very close to 150, we always round up when it comes to minimum sample size problems like this.
Why up? Because it ensures we clear the hurdle.
The larger n gets, the smaller E gets.
Intuitively this fits with the idea that a larger sample is more representative; hence it would more accurately capture the parameter we're after (leading to reduced error).
Let's say we go with n = 150
Calculating the margin of error gets us
E = z*sqrt(phat*(1-phat)/n)
E = 1.96*sqrt(0.5*(1-0.5)/150)
E = 0.08001666493091
This error is a bit too high, as it is larger than 0.08
Though you could argue that rounding it to two decimal places gets us E = 0.08 just fine.
Standard practice is to round up to ensure E is under 0.08
So if we go for n = 151 then we have
E = z*sqrt(phat*(1-phat)/n)
E = 1.96*sqrt(0.5*(1-0.5)/151)
E = 0.07975126895958
We've cleared the hurdle and E is now smaller than 0.08
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