Problem 13SE

A 99% confidence interval for a population mean based on a sample of size 64 is computed to be (16.3, 18.7). How large a sample is needed so that a 99% confidence interval will specify the mean to within ±1.0?

Answer:

Step 1 of 1:

In this question, we are asked to find the number of samples needed so that a confidence interval will specify the mean to within .

A confidence interval for a population mean based on a sample of size 64 is computed to be .

Let be a large sample of size from a population with mean and standard deviation

then a level confidence interval for is

……………..(1)

Given level of confidence , sample size , and confidence interval .

An upper confidence bound = and lower confidence bound

We know from equation (1),

Mean width = confidence interval

width = an upper confidence bound

width = lower confidence bound

width = 18.7 …………(2)

width = 16.3 ……………….(3)

On solving equation (2) and (3),

, and width

width

Since we want 99% confidence interval, then a level is

and

Then for is , hence

Now we will find out the number of samples for given mean to be within .

width

Hence the number of sample for given mean to be within is 92.