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 (y.y2) denote the outcomes for rolling the die twice.

(a) Enumerate the 36 possible (y1,32) pairs (e.g.. (2, 1) represents a 2 followed by a 1).

(b) Treating the 36 pairs as equally likely, con- struct 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 distri- bution 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 mid- dle than at the minimum and maximum values. (Hint: Note there are many more (y, y2) pairs that have a sample mean near the middle than near the minimum or maximum.)