Exercise 1 A serving of breakfast cereal has a sugar content that is well approximated by a Normal distributed random variable X with mean 13 g and variance: 1.3 2 g 2 . We can consider each serving as an independent and identical draw from X.

a. In what percent of servings will the sugar content be above 13.3 g?

b. What is the probability that a randomly chosen serving will have a sugar content between 13.877 and 12.123? What do we call the difference: 13.877-12.123=1.754?

c. Calculate the probability that in 6 servings, only 1 has a sugar content below 13 g.

d. Describe the sampling distribution for the mean sugar content of 6 servings X.

e. What is the interquartile range of the sampling distribution for the sample mean X when n=6? Is that value larger or smaller than the IQR implied in part b? Why does the relative sizes of the IQRs make sense?

f. What is the probability that the mean sugar content in 6 servings is more than 13.3 g ?

g. Is it more or less likely that the mean sugar content is above 13.3 g in 10 servings or 6 servings (as computed in f)? How can you explain your answer without even computing the new probability?

h. Suppose each cereal box of this type contains 10 servings and consider the total sugar content in each box as a sum of 10 iid random draws from X ? N(13, 1.3 2 ). If you were to eat a whole box of cereal, above what total sugar content would you consume with 95% probability? Show and briefly explain your calculations.

3. Retail stores experience their heaviest volume of transactions that include returns on December 26th and December 27th each year. The distribution for the Number of Items Returned (X) by Macy's customers who do a return transaction on those days last year is given in the table below. It has mean: = 2.61 and variance ? 2 ? 1.80.

Number of Items Returned in Transaction (X) Probability x=1 0.25 x=2 0.28 X=3 0.20 X=4 0.17 X=5 0.08 X=6 0.02