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 20% 14% 21% 17% 18% 10% (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. No. The events are indistinct and the probabilities sum to more than 1. No. The events are indistinct and the probabilities sum to 1. Yes. The events are distinct and the probabilities sum to 1. (b) Use a histogram to graph the probability distribution of part (a). Maple Generated Plot Maple Generated Plot Maple Generated Plot Maple Generated Plot (c) Compute the expected income μ of a super shopper. (Round your answer to two decimal places.) μ = thousands of dollars (d) Compute the standard deviation σ for the income of super shoppers. (Round your answer to two decimal places.) σ = thousands of dollars

Answer 1

Income range	Midpoint x	Percent of super shoppers	P(x=x)
5-15	10	20%	0.20
15-25	20	14%	0.14
25-35	30	21%	0.21
35-45	40	17%	0.17
45-55	50	18%	0.18
55 or more	60	10%	0.10
sum			1

(a) Using the income midpoints x and the percent of super shoppers, do we have a valid probability distribution?

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

#Ansb) histogram

c)

  

µ = 10*0.20+20*0.14+30*0.21+40*0.17+50*0.28+60*0.10 = 32.9

µ =32.9

d)

σ=sqrt(∑x2⋅p(x)−μ2)

∑x2⋅p(x)=1347

σ=sqrt(∑x2⋅p(x)−μ2)=sqrt(1347−(32.9)2)=16.27

#σ =16.27