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?
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
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
Since we want 99% confidence interval, then a level is
Then for is , hence
Now we will find out the number of samples for given mean to be within .
Hence the number of sample for given mean to be within is 92.