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	22%	15%	20%	16%	19%	8%

(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. Yes. The events are indistinct and the probabilities sum to less than 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. No. The events are indistinct and the probabilities sum to 1.

(b)

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

    

(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 = thousands of dollars

(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 = thousands of dollars

Answer 1

a)

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

b)

c)

X	P(X)	X*P(X)
10	0.2200	2.200
20	0.1500	3.000
30	0.2000	6.000
40	0.1600	6.400
50	0.1900	9.500
60	0.0800	4.800

mean = E[X] = Σx*P(X) =            31.9

d)

X	P(X)	X*P(X)	X² * P(X)
10	0.2200	2.200	22.000
20	0.1500	3.000	60.000
30	0.2000	6.000	180.000
40	0.1600	6.400	256.000
50	0.1900	9.500	475.000
60	0.0800	4.800	288.000

mean = E[X] = Σx*P(X) =            31.9000
          
E [ X² ] = ΣX² * P(X) =            1281.0000
          
variance = E[ X² ] - (E[ X ])² =            263.3900
          
std dev = √(variance) =            16.23

THANKS

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