Chapter 8.1, Problem 7E

Consider the following population: {1, 2, 3, 4}. For this population the mean is

$\mu =\frac{1+2+3+4}{4}=2.5$

Suppose that a random sample of size 2 is to be selected without replacement from this population. There are 12 possible samples (assuming that the order in which observations are selected is taken into account):

1. a. Calculate the sample mean for each of the 12 possible samples.
2. b. Use the sample means to construct the sampling distribution of x̄. Display the sampling distribution as a density histogram.
3. c. Suppose that a random sample of size 2 is to be selected, but this time sampling will be done with replacement. Using a method similar to that of Part (a), construct the sampling distribution of x̄. (Hint: There are 16 different possible samples in this case.)
4. d. In what ways are the two sampling distributions of Parts (b) and (c) similar? In what ways are they different?