Continuous Random Variables
Continuous Random Variables
Designing Sports Cars
Jeffrey L. Rummel
Emory University
Goizueta Business School
MBA15 Statistics Foundations
Jeffrey L. Rummel
MBA15 Statistics Foundations
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Continuous Random Variables
Application of continuous random variables
Fitting into your sports car
Daniel Craig tooled around as James Bond in his Aston Martin,
and luckily for the movie makers, he is only 5 feet 10 inches tall.
Thats because in order to look and drive the way they do, many
sports cars have only limited headroom, and a taller actor in a
hardtop DBS that is only 50 inches tall might not have fit. In
other movies with exotic sports cars, you often see convertibles
with the top down through the entire movie.
As it turns out, most exotics (Ferrari, Lamborghini, Lotus) have
very little headroom. In fact, anyone taller than about 6 feet will
find it difficult to fit without touching the roof with their head.
Jeffrey L. Rummel
MBA15 Statistics Foundations
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Continuous Random Variables
Application of continuous random variables
Fitting into your sports car
I
Assume that the total population has a height that follows a
normal distribution
I
Assume that the population has a mean height of 5 feet 7
inches
I
Assume that the population has a standard deviation of 4.2
inches
Jeffrey L. Rummel
MBA15 Statistics Foundations
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Continuous Random Variables
Application of continuous random variables
Questions
1. What is the probability that someone who decides to buy an
exotic car will be too tall to fit?
I
Find the z value for someone 6 feet tall
I
z = (72 − 67)/4.2 = 1.1905
I
Look z value up (or use =NORMDIST()), find that
F (z) = .8831
I
Too tall is 1 − F (z) so about 12% of people will not fit.
Jeffrey L. Rummel
MBA15 Statistics Foundations
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Continuous Random Variables
Application of continuous random variables
Questions
2. How unusual is a person that is 6 feet 6 inches tall?
I
The z value this time is (78 − 67)/4.2 = 2.6190
I
Can find that F (z) = .9956
I
Only 0.4% people are taller than 6 feet 6 inches tall
Jeffrey L. Rummel
MBA15 Statistics Foundations
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Continuous Random Variables
Application of continuous random variables
Questions
3. Suppose drivers from 6 feet to 6 feet 2 inches can drive by
leaning their head; what additional percentage of the population
could squeeze in that way?
I
For 6 feet, z = 1.1905 and F (z) = .8831
I
For 6 feet 2 inches, z = (74 − 67)/4.2 = 1.6667 and
F (z) = .9522
I
So we add about 7% of the potential customers if leaning is
acceptable
Jeffrey L. Rummel
MBA15 Statistics Foundations
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Continuous Random Variables
Application of continuous random variables
Questions
4. For how tall a person should we design our car so that all but
the 4% tallest people would fit?
I
Want 96% to fit, so find .96 in the z table, or use =NORMINV()
I
z = 1.7507
I
Height can be solved for: 1.7507 = (X − 67)/4.2
I
4% of people are taller than 6 feet 2.35 inches tall
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MBA15 Statistics Foundations
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Continuous Random Variables
Application of continuous random variables
Questions
5. We have an additional problem: in order to fit taller people, the
design can affect people at the other end of the distribution, and if
the driver is shorter than 5 feet 5 inches they will not be able to
see over the dashboard. What is the probability that a prospective
buyer is too short?
I
For 5 feet 5 inches, z = (65 − 67)/4.2 = −.4762 and
F (z) = .3170
I
So about 32% of customers will be too short
Jeffrey L. Rummel
MBA15 Statistics Foundations
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Continuous Random Variables
Application of continuous random variables
Questions
5. (continued) If drivers from 5 feet 5 inches to 5 feet 1 inches can
“scoot up” to see, how many (i.e. what additional percentage of
the population) could get by?
I
For 5 feet 5 inches, z = (65 − 67)/4.2 = −.4762 and
F (z) = .3170
I
So about 32% of customers will be too short
I
For 5 feet 1 inch, z = (61 − 67)/4.2 = −1.4286 and
F (z) = .0766
I
About 8% of customers will not be able to see even if they
scoot up
I
If scooting is acceptable, then we add about 24% of potential
customers
Jeffrey L. Rummel
MBA15 Statistics Foundations
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Continuous Random Variables
Application of continuous random variables
Questions
6. How much does our seat need to rise up to get all but 1% of
the population?
I
Look up 1% in the z table and find that z = −2.3263
I
Solve for the height: −2.3263 = (X − 67)/4.2
I
Customers less than 4 feet 9.23 inches are about 1% of the
population
I
If 5 feet 1 inch customers can see, the seat needs to rise
another 3.77 inches
I
To eliminate scooting, the seat needs to rise another 7.77
inches
Jeffrey L. Rummel
MBA15 Statistics Foundations
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Continuous Random Variables
Application of continuous random variables
Questions
7. What is the probability that a potential customer will walk in
and have no problems fitting in the car (in other words, no
scooting up or leaning over)?
I
Want from 5 feet 5 inches to 6 feet
I
The two z values are 1.1905 and −0.4762
I
The two F (z) values are .8831 and .3170
I
This means almost 57% of the population will have no
problem
Jeffrey L. Rummel
MBA15 Statistics Foundations
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Continuous Random Variables
Application of continuous random variables
Questions
7. (continued) What is the probability that a potential customer
will walk in and have no problems fitting in the car ( they would be
willing to scoot up or lean over)?
I
Want from 5 feet 1 inches to 6 feet 2 inches
I
If we allow leaning and scooting, now the z values are 1.6667
and −1.4286
I
The two F (z) values are .9522 and .0766
I
This means almost 88% of the population can use our car
Jeffrey L. Rummel
MBA15 Statistics Foundations
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