Problem 5E

A random sample of 70 observations from a normally distributed population possesses a sample mean equal to 26.2 and a sample standard deviation equal to 4.1.

a. Find an approximate 95% confidence interval for μ.

b. What do you mean when you say that a confidence coefficient is .95?

c. Find an approximate 99% confidence interval for μ.

d. What happens to the width of a confidence interval as the value of the confidence coefficient is increased while the sample size is held fixed?

e. Would your confidence intervals of parts a and c be valid if the distribution of the original population was not normal? Explain.

Answer

Step 1 of 5

(a)

Given a random sample of observations which follows a normally distributed population.

Given sample mean and sample standard deviation

We are asked to find an approximate confidence interval for

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

then a level confidence interval for is

……………..(1)

Given sample mean , sample standard deviation and

Since we want 95% confidence interval, then a level is

and

Then for is , hence

Hence 95% confidence interval for is

Hence a 95% confidence interval for is .