Question

A simple random sample of size n is drawn from a population that is normally distributed. The sample​ mean, x over bar​, is found to be 106​, and the sample standard​ deviation, s, is found to be 10. ​(a) Construct a 96​% confidence interval about mu if the sample​ size, n, is 24. ​(b) Construct a 96​% confidence interval about mu if the sample​ size, n, is 19. ​(c) Construct an 80​% confidence interval about mu if the sample​ size, n, is 24. ​(d) Could we have computed the confidence intervals in parts​ (a)-(c) if the population had not been normally​ distributed?

Answer 1

Here sample mean , sample standard deviation s=10

Since here we don't have population standard deviation we use Students's t distribution to get the confidence intervals.

Here % confidence interval for population mean is

where n= sample size, is the t-value with v= n-1 degrees of freedom. You can find this formula in any applied statistics book.

a) here n=24    v=n-1=23 2.177

Therefore

b)n=19 v=n-1=18 2.214

c)n=24 v=n-1=23    1.319

d) Here for calculating the confidence intervals, the assumption is that the population is normally distributed. So, if the population is not normally distributed we can not use the above formula and we can not compute confidence intervals.