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	10%	43%	20%	10%	17%

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.

No. The events are distinct and the probabilities sum to 1.Yes. The events are indistinct and the probabilities sum to less than 1.     Yes. The events are distinct and the probabilities do not 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.

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

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

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

Answer 1

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

b)

c)

X	P(X)	X*P(X)	X² * P(X)
23	0.1	2.3	52.900
34	0.43	14.62	497.080
45	0.2	9	405.0000
56	0.1	5.6	313.6000
67	0.17	11.3900	763.1300

expected age=µ= mean = E[X] = Σx*P(X) =            42.91 years

d)

E [ X² ] = ΣX² * P(X) =            2031.7100
          
variance = E[ X² ] - (E[ X ])² =            190.4419
          
std dev = √(variance) =            13.80 yr