Question

A 95​% confidence interval of 17.6 months to 49.2 months has been found for the mean duration of​ imprisonment, mu​, of political prisoners of a certain country with chronic PTSD. a. Determine the margin of​ error, E. b. Explain the meaning of E in this context in terms of the accuracy of the estimate. c. Find the sample size required to have a margin of error of 11 months and a 99​% confidence level.​ (Use sigmaequals45 ​months.) d. Find a 99​% confidence interval for the mean duration of​ imprisonment, mu​, if a sample of the size determined in part​ (c) has a mean of 36.5 months.

Answer 1

a)
`E = (49.2-17.6)/(2)= 15.8`

b) For this case we have 95% of confidence that the true mean would be between
`\pm 15.8` units respect the true mean.

c)
`n=((2.58(45))/(11))^2 =111.39 \approx 112`

So the answer for this case would be n=112

d)
`36.5-2.58(45)/(√(112))=25.53`

`36.5+2.58(45)/(√(112))=47.47`

Explanation:

Part a

For this case we know that the cinfidence interval for the true mean is given by:

`\bar X \pm E`

Where E represent the margin of error. For this case we have the confidence interval at 95% of confidence and we can estimate the margin of error like this:

`E = (49.2-17.6)/(2)= 15.8`

Part b

For this case we have 95% of confidence that the true mean would be between
`\pm 15.8` units respect the true mean.

Part c

The margin of error is given by this formula:

`ME=z_(\alpha/2)(\sigma)/(√(n))` (a)

And on this case we have that ME =11 and we are interested in order to find the value of n, if we solve n from equation (a) we got:

`n=((z_(\alpha/2) \sigma)/(ME))^2` (b)

The critical value for 99% of confidence interval now can be founded using the normal distribution. The critical value would be
`z_(\alpha/2)=2.58`, replacing into formula (b) we got:

`n=((2.58(45))/(11))^2 =111.39 \approx 112`

So the answer for this case would be n=112

Part d

The confidence interval for the mean is given by the following formula:

`\bar X \pm z_(\alpha/2)(\sigma)/(√(n))` (1)

Replcaing the info given we got:

`36.5-2.58(45)/(√(112))=25.53`

`36.5+2.58(45)/(√(112))=47.47`