Question

In the library on a university campus, there is a sign in the elevator that indicates a limit of 16 persons. Furthermore, there is a weight limit of 2500 lb. Assume that the average weight of students, faculty, and staff on campus is 151 lb, that the standard deviation is 25 lb, and that the distribution of weights of individuals on campus is approximately normal. If a random sample of 16 persons from the campus is to be taken:

(a) What is the expected value of the sample mean of their weights? (The x in ?x has some horizontal line over it)
?x = ________ lb

(b) What is the standard deviation of the sampling distribution of the sample mean weight? (Round your answer to two decimal places.) (The x in ?x has some horizontal line over it)
?x = ________ lb

Answer 1

a) 151lb.

b) 6.25 lb

Explanation:

The Central Limit Theorem estabilishes that, for a random variable X, with mean
`\mu` and standard deviation
`\sigma`, a large sample size can be approximated to a normal distribution with mean
`\mu` and standard deviation
`(\sigma)/(√(n))`.

In this problem, we have that:

`\mu = 151, \sigma = 25, n = 16`

So

a) The expected value of the sample mean of the weights is 151 lb.

(b) What is the standard deviation of the sampling distribution of the sample mean weight?

This is
`s = (\sigma)/(√(n)) = (25)/(√(16)) = 6.25`