Question

A group of 501 students took the college entrance exam, and had a mean score of
372 and a standard deviation of 27. Suppose a random sample of 81 students is
selected.
A. If the sampled population is not very large, we need to make an adjustment in
the calculation of standard error. Explain why.
B. Under what condition do we need to make an adjustment in the calculation of
standard error? Check if this condition holds.
C. What is the distribution of sample mean after the adjustment?
D. What is the probability that the sample mean score is greater than 365?

Answer 1

1:

Here sampling is done without replacement and population is not very large from sample so we need to use finite population correction factor. It is necessary because in these conditions central limit theorem cannot be applied.

Applying fpc, will cover the difference of sampling with replacement and sampling without replacement.

2:

Since

So it is necessary to apply the finite population correction factor.

C:

Here finite population correction factor is

The distribution of sample mean after the adjustment will be approximately normal with mean and standard deviation as follows:

D:

The z-score for is

Using z table, the probability that the sample mean score is greater than 365 is