Random Variables Classification and Market Age

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1.) Random Variables: Classification- Which of the following are continuous variables and which are discrete?
a.) Speed of an airplane
b.) Age of college professor chosen at random
c.) Number of books in the college bookstore
d.) Weight of a football player chosen at random
e.) Number of lightning strikes in Rock Mountain National Park on a given day

Solution:
a.) Speed of an airplane = Continuous Variable
b.) Age of college professor chosen at random = Discrete Variable
c.) Number of books in the college bookstore = Discrete Variable
d.) Weight of a football player chosen at random = Discrete
e.) Number of lightning strikes in Rock Mountain National Park on a given day = Continuous Variable

2.) Marketing Age- 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. The following table is based on information taken from trends in the United States (Food Marketing Institute, Washington, D.C.).

Age range, years 18-29 29-39 40-50 51-61 62 and over
Midpoint x 23 34 45 56 67
Percent of super shoppers 7% 44% 24% 14% 11%

For the 62 and over age group, use the midpoint 67 years.
a.) Using the age midpoints x and the percentages of super shoppers, do we have a valid probability distribution? Explain.
b.) Use a histogram to graph the probability distribution of part (a)
c.) Compute the standard deviation ₀ for ages of super shoppers.

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

We have a probability distribution as Binomial probability distribution

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

X p(x)
23 0.07
34 0.44
45 0.24
56 0.14
67 0.11

c.) Compute the standard deviation ₀ for ages of super shoppers.

σ = √ Σ (x - µ)² P( x ) =

X p(x) X*p(X) (X-µ)^2 ((X-µ)^2)*p(X)
23 0.07 1.61 383.3764 26.8363
34 0.44 14.96 73.6164 32.3912
45 0.24 10.8 5.8564 1.4055
56 0.14 7.84 180.0964 25.213
67 0.11 7.37 596.3364 65.5970
74.5 0.091 6.7795 438.4836 39.9020076
84.5 0.018 1.521 957.2836 17.2311048
42.58 151.443

=151.443
So, the standard deviation σ is

3.) History: Florence Nightingale- What was the age distribution of nurses in Great Britain at the time of Florence Nightingale? Thanks to Florence Nightingale and the British census of 1851, we have the following information (based on data from the classic text Notes of Nursing, by Florence Nightingale). Note: In 1851 there was 25,466 nurses in Great Britain. Furthermore, Nightingale made a strict distinction between nurses and domestic servants.

Age range (yr) 20-29 30-39 40-49 50-59 60-69 70-79 80+
Midpoint x 24.5 34.5 44.5 54.5 64.5 74.5 84.5
Percent of nurses 5.7% 9.7% 19.5% 29.2% 25.0% 9.1% 1.8%

a.) Using the age midpoints x and the percentage of nurses, do we have a valid probability distribution? Explain.
b.) Use a histogram to graph the probability distribution of part (a).
c.) Find the probability that a British nurse selected at random in 1851 would be 60 years of age or older.
d.) Compute the expected age µ of a British nurse contemporary to Florence Nightingale.
e.) Compute the standard deviation ₀ for ages of nurses shown in the distribution.
Solution:
a). Using the age midpoints x and the percentages of nurses, do we have a valid probability distribution? Explain

We have a probability distribution as Binomial probability distribution

b). Use a histogram to graph the probability distribution of part (a).
X p(x)
24.5 0.057
34.5 0.097
44.5 0.195
54.5 0.292
64.5 0.25
74.5 0.091
84.5 0.018

c). Find the probability that a British nurse selected at random in 1851 would be 60 years of age or older
The probability that a British nurse selected at random in 1851 would be 60 years of age or older is = P(X=64.5)+ P(X=74.5)+ P(X=84.5)=0.25+0.091+0.018=0.359

d). Compute the expected age µ of a British nurse contemporary to Florence Nightingale.
µ = ΣxP(x); µ = x
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