Chapter PV, Problem 11RE

(a)

To determine

To find out what is the expected shape of the sample’s distribution as our sample size increases.

Answer to Problem 11RE

It is bimodal.

Explanation of Solution

It is given in the question that we are sampling randomly from a distribution known to be bimodal. Thus, as our sample size increases then the expected shape of the sample’s distribution is stated as being bimodal, which has two modes in the sampling distribution.

(b)

To determine

To find out what is the expected value of our sample’s mean and explain does this size of the sample matter.

Answer to Problem 11RE

  $E(x)=\mu$ and the sample size does not matter.

Explanation of Solution

It is given in the question that we are sampling randomly from a distribution known to be bimodal. Thus, as we know that the expected value of our sample’s mean will be:

  $E(x)=\mu$

That is our population mean and is independent of n . Also, the size of the sample does not matter as the sample mean is independent of sample size n .

(c)

To determine

To explain how is the variability of sample means related to the standard deviation of the population and does the size of the sample matter.

Answer to Problem 11RE

Yes, the sample size matter.

Explanation of Solution

It is given in the question that we are sampling randomly from a distribution known to be bimodal. Thus, as we know that the expected value of our sample’s mean will be:

  $E(x)=\mu$

Thus, the variability of sample means is related to the standard deviation of the population as the standard deviation $SD\propto \frac{1}{\sqrt{n}}$ decreases with n , meaning that the variance of $\mu$ decreases with increasing n . Thus, the size of the sample matter in the standard deviation.

(d)

To determine

To explain how is the shape of the sampling distribution model affected by the sample size.

Explanation of Solution

It is given in the question that we are sampling randomly from a distribution known to be bimodal. Thus, as we know that the expected value of our sample’s mean will be:

  $E(x)=\mu$

Thus, the shape of the sampling distribution model is affected by the sample size as by the central limit theorem, the sampling distribution of any mean or proportion is normal such that we expect an increasingly normal model with increasing n .