Question

What is the age distribution of promotion-sensitive shoppers? A supermarket super shopper is defined as a shopper for whom at least 70% of the items purchased were on sale or purchased with a coupon. Age range, years 18-28 29-39 40-50 51-61 62 and over Midpoint x 23 34 45 56 67 Percent of super shoppers 5% 48% 24% 10% 13% For the 62-and-over group, use the midpoint 67 years.

(a) Using the age midpoints x and the percentage of super shoppers, do we have a valid probability distribution? Explain. Yes. The events are indistinct and the probabilities sum to less than 1. No. The events are distinct and the probabilities sum to 1. No. The events are indistinct and the probabilities sum to more than 1. Yes. The events are distinct and the probabilities sum to 1. Yes. The events are distinct and the probabilities do not sum to 1.

(b) Use a histogram to graph the probability distribution of part (a). (c) Compute the expected age ? of a super shopper. (Round your answer to two decimal places.) ? = yr (d) Compute the standard deviation ? for ages of super shoppers. (Round your answer to two decimal places.) ? = yr

Answer 1

A) Using the age midpoints x and the percentage of super shoppers, do we have a valid probability distribution? Explain.

Yes. The events are indistinct and the probabilities sum to less than 1

No. The events are distinct and the probabilities sum to 1

No. The events are indistinct and the probabilities sum to more than 1

Yes. The events are distinct and the probabilities sum to 1.

Yes. The events are distinct and the probabilities do not sum to 1.

Sum of P(X) = 0.05+0.48+0.24+0.1+0.13 = 1

Thus they have valid probability distribution where each event is distinct and P(X) sums to 1.

(b) Use a histogram to graph the probability distribution of part (a).

Age range	X	P(X)
18-28	23	0.05
29-39	34	0.48
40-50	45	0.24
51-61	56	0.1
> 62	67	0.13
	SUM	1

(c) Compute the expected age ? of a super shopper. (Round your answer to two decimal places.) ? = yr

(d) Compute the standard deviation ? for ages of super shoppers. (Round your answer to two decimal places.) ? = yr

The expected age and standard deviation of the age of super shoppers is calculated using the following formula

Age range	X	P(X)	X.P(X)	(X-mu)^2*P(X)
18-28	23	0.05	1.15	19.16882
29-39	34	0.48	16.32	35.335872
40-50	45	0.24	10.8	1.405536
51-61	56	0.1	5.6	18.00964
> 62	67	0.13	8.71	77.523732
	SUM	1	42.58	151.4436

Expected Age = 42.58 yr

Standard deviation of age = sqrt(151.4436) = 12.31 yr