4.28. The probability distribution associated with the outcome of rolling a balanced die has probability 1/6 attached to each integer, {1,2,3,4,5,6}. Let

(yi^yi) denote the outcomes for rolling the die twice.

(a) Enumerate the 36 possible (yi,^) pairs (e.g.,

(2.1) represents a 2 followed by a 1).

(b) Treating the 36 pairs as equally likely, construct the sampling distribution for the sample mean y of the two numbers rolled.

(c) Construct a histogram of the (i) probability distribution for each roll, (ii) sampling distribution of y in (b). Describe their shapes.

(d) What are the means of the two distributions in (c)? Why are they the same?

(e) Explain why the sampling distribution of y has relatively more probability near the middle than atthe minimum and maximum values.

{Hint: Note there are many more (yi,^) pairs that have a sample mean near the middle than near the minimum or maximum.)