Question

Find probability for sampling distribution using Central Limit Theorem

The mean room and board expense per year at four-year college is $7540.

You randomly selected 9 four-year colleges. What is the probability that the mean room and board is less than $7800? Assume that the room and board expenses are normally distributed with a standard deviation of $1245.

Answer 1

CLT:
Central limit theorem states that if samples of size n, (n ≥ 30) are drawn from any population with mean μ and standard deviation σ, the sample mean will be approximately distributed according to a normal distribution with μ_x = μ and standard deviation σ_x = σ / √n.

Probability that the mean room and board is less than $7800 for a smaple of 9 four year colleges.

Although the sample size is less than 30 , CLT can still be used because the random variable representing the yearly expense is assumed to be normally distirbuted.

Given population mean μ = 7540 and population standard deviation σ_x = 1245
Let μ_x represent the mean of sample.
Hence P(μ_x < 7800) = P( (μ_x - μ) / ( σ / √n) < (7800 - 7540) / (1245/ √9) )
z = 260/415 ~ 0.63

For the probability that the mean room and board is less than $7800, the area to the left of the z-score is needed, thus, using the standard normal tables, the desired probability is 0.7357