Question

In a large population of prisoners, the mean IQ is 95 with a standard deviation of 15. 250 adults from this population are randomly selected for a survey of attitudes toward crime.
a. What is the mean of this sampling distribution (all samples of n = 250)?
b. What is the standard deviation of the sampling distribution?
c. What is the shape of sampling distribution?
d. If an individual prisoner has an IQ of 90, is she a likely or unlikely example?
e. If you change the sample size to 100, will the mean of the sampling distribution change?

Answer 1

a) The mean of sampling distribution with sample size of 250 will also be the same as of the population as 95 .

Because of central limit theorem for normal population which states that the mean of a sample taken from a normally distributed population will be equal to the population mean.

b) The standard deviation of the sample with sample size is calculated as

$S=\frac{\sigma}{\sqrt{n}}=\frac{15}{\sqrt{250}}=0.949$

By the central limit theorem.

c) Since the population is normally distributed and the sample is taken from. Normal population and also the sample size is more than 30, then we can assume the shape of distribution as symmetrical and bell shaped.

d) Yes it is very likely to have a IQ of an individual of 90.

e) According to the central limit theorem whatever is the sample size , of the sample is drawn from a normally distributed population then the mean of sample will be equal to the population mean. Hence here sample mean= 95 and it remains the same.