An investigator computes a 95% confidence interval for a population mean on the basis of a sample of size 75 if she wishes to compute a 95% confidence interval that is half as wide, how large a sample does she need?

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link223mh

Answered question

2022-10-31

An investigator computes a 95 % confidence interval for a population mean on the basis of a sample of size 75 if she wishes to compute a 95 % confidence interval that is half as wide, how large a sample does she need?

Answer & Explanation

imperiablogyy

imperiablogyy

Beginner2022-11-01Added 13 answers

Step 1
The width of a confidence interval is
C n
where C is some number that is proportional to the standard deviation and also has a factor (like 1.96) that is determined by what type of confidence interval and what percentile. The only important thing here is how it depends on the sample size.
So the width of your first interval is
w 1 = C 75
since the sample size is 75. If you make your second sample size n 2 rather than 75, the width will be
w 2 = C n 2 .
We want the second to be half as wide, so
w 2 = 1 2 w 1 .
Plugging in for w 1 and w 2 , we want
C n 2 = 1 2 C 75 .Then we solve for n 2 :
n 2 = 2 75 n 2 = 4 75 = 300 .

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