SOLUTION: The College of Podiatrists states that 48% of women wear shoes that are too small for their feet. A researcher wants to be 98% confident that this proportion is within 0.05 of t
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Question 1204120: The College of Podiatrists states that 48% of women wear shoes that are too small for their feet. A researcher wants to be 98% confident that this proportion is within 0.05 of the true proportion. How large a sample is necessary?
Which answer is correct?
718, 413, 540, or 286? Found 2 solutions by Theo, MathLover1:Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website! population mean proportion is .48.
98% confidence interval with two tails requires critical z-score of plus or minus 2.326347.
margin of error is plus or minus .05.
z-score formula is z = (x-m)/s
high side z-score = 2.326347.
(x-m) is equal to .05 which is the high side margin of error.
s = standard error = sqrt(.48 * .52 / n) = sqrt(.2496/n)
n is the sample size.
z-score formula becomes 2.326347 = .05 / sqrt(.2496/n)
mutiply both sides of the equation by sqrt(.2496/n) and divide both sides of the formula by 2.326347 to get:
sqrt(.2496/n) = .05 / 2.326347
square both sides of the equation to get:
.2496/n = (.05/2.326347)^2
solve for n to get:
n = .2496 / (.05/2.326347)^2 = 540.323134.
standard error becomes equal to sqrt(.2496/540.323134) = .021493.
z-score formula becomees:
2.326347 = (x-m) / .021493.
solve for (x-m) to get:
(x-m) = 2.326347 * .021493 = .050000.
that's your margin of error.
calculator says that 98% confidence interval is between .43 and .53.
.48 minus .43 = .05
.53 minus .48 = .05
margin of error is .05 as desired.
here are the results from that calculator.
your sample size needs to be an integer, so round 540.323134 to 541 and that's your solution.
you will get a margin of error slightly less than .05 when you do that.
your revised standared error will be equal to sqrt(.2496/541) = .021479.
your margin of error becomes .021479 * 2.326347 = .049970 which is slightly less than .05 = within .05, as required.
The sample size () is calculated according to the formula:
Where: for a confidence level (α) of = proportion (expressed as a decimal), = margin of error.
