Question

What is the income distribution of super 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. In the following table, income units are in thousands of dollars, and each interval goes up to but does not include the given high value. The midpoints are given to the nearest thousand dollars.
Income range   5-15   15-25   25-35   35-45   45-55   55 or more
Midpoint x   10   20   30   40   50   60
Percent of super shoppers   21%   15%   22%   16%   20%   6%
(a) Using the income midpoints x and the percent of super shoppers, do we have a valid probability distribution? Explain.
Yes. The events are distinct and the probabilities do not sum to 1.
No. The events are indistinct and the probabilities sum to 1.
Yes. 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 indistinct and the probabilities sum to less than 1.

(c) Compute the expected income μ of a super shopper (in thousands of dollars). (Enter a number. Round your answer to two decimal places.)
μ =

(d) Compute the standard deviation σ for the income of super shoppers (in thousands of dollars). (Enter a number. Round your answer to two decimal places.)
σ =

Answer 1

a)

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

c)

The expected income is obtained using the formula,

From the data values,

Midpoint, X	Probability, P(X)	X*P(X)
10	0.21	2.1
20	0.15	3
30	0.22	6.6
40	0.16	6.4
50	0.2	10
60	0.06	3.6
	Sum	31.7

d)

The standard deviation for income is obtained using the formula,

Midpoint, X	Probability, P(X)	x^2*P(X)	X*P(X)
10	0.21	100	21
20	0.15	400	60
30	0.22	900	198
40	0.16	1600	256
50	0.2	2500	500
60	0.06	3600	216
		Sum	1251