QA 233 (Basic Business Statistics)
Solutions to Practice Problem Set VI
Marsha
Mallo is a Quality Assurance Technician for Pillar Foods, Inc., producers of
Choco-Poofies. Ms. Mallo is primarily responsible for ascertaining that the
popular children’s breakfast cereal contains the desired quantity of chocolate
candy pieces. The chocolate candy pieces are far more expensive than the oat
cereal pieces they are mixed with to produce Choco-Poofies, so Pillar Foods does
not want the amount of chocolate candies in each box of Choco-Poofies to be too
high (i.e., the producer’s risk).
On the other hand, the chocolate candies are what makes Choco-Poofies popular
with children, so Pillar Foods also does not want the amount of chocolate
candies in each box of Choco-Poofies to be too low (i.e., the consumer’s
risk). In response to these two concerns, Pillar Foods has decided that
their most popular sized box of Choco-Poofies (the thirty-six ounce economy
size) should contain a mean of twelve ounces of chocolate pieces. Furthermore,
they have also decided that the variance in the amount of chocolate pieces
contained in a thirty-six ounce box of Choco-Poofies should be 1.96. In
reviewing the historical production data, Marsha has found that the amount of
chocolate candies in thirty-six ounce boxes of Choco-Poofies to be normally
distributed.
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Suppose
a financial analyst at Pillar Foods has determined that Pillar will loose
money on a thirty-six ounce box of Choco-Poofies if it contains more than
fourteen ounces of chocolate candies. Help Marsha determine the
probability that a randomly selected thirty-six ounce box of Choco-Poofies
will expose Pillar to the producer’s risk. In other words, what is the
probability that Pillar will loose money on a box of Choco-Poofies due to
overfill of chocolate candies if the production process is working at the
target levels (a mean of twelve ounces and a variance of 1.96) set by
Pillar?
Our
random variable x is the amount (in ounces) of chocolate candies in a
thirty-six ounce box of Choco-Poofies. If the production process is working
at the target levels set by Pillar, then x is normally distributed with a
mean of 12.0 ounces and a standard deviation of 1.40. We can write this
question as
P(x>14)
which
we can rewrite as
P(12£x£¥)-P(12£x£14)
In
terms of the standard normal probability distribution this is
which
we can solve using the standard normal probability table:
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Suppose
a marketing analyst at Pillar Foods has determined that Pillar risks
loosing a youthful customer of Choco-Poofies if a thirty-six ounce box
contains less than eleven ounces of chocolate candies. Help Marsha
determine the probability that a randomly selected thirty-six ounce box of
Choco-Poofies will expose Pillar to the consumer’s risk. In other words,
what is the probability that a thirty-six ounce box of Choco-Poofies will
put Pillar at risk of loosing a customer if the production process is
working at the target levels (a mean of twelve ounces and a variance of
1.96) set by Pillar?
Again,
our random variable x is the amount (in ounces) of chocolate candies in a
thirty-six ounce box of Choco-Poofies. If the production process is working
at the target levels set by Pillar, then x is normally distributed with a
mean of 12.0 ounces and a standard deviation of 1.40. We can write this
question as
P(x<11)
which
we can rewrite as
P(-¥£x£12)-P(11£x£12)
In
terms of the standard normal probability distribution this is
which
we can solve using the standard normal probability table:
-
If
the production process is working at the target levels (a mean of twelve
ounces and a variance of 1.96) set by Pillar, what is the probability that
a randomly selected thirty-six ounce box of Choco-Poofies will expose
Pillar to either the
producer’s risk or the consumer’s risk? What is the probability that a
randomly selected thirty-six ounce box of Choco-Poofies will expose Pillar
to neither the producer’s
risk or the consumer’s risk?
We
can use our previous results to determine that the probability that a
randomly selected thirty-six ounce box of Choco-Poofies will expose Pillar
to neither
the producer’s risk or the consumer’s risk is
P(11£x£14)=P(11£x£12)+P(12£x£14)=0.2612+0.4236=0.6848
We
can use this result to easily find Choco-Poofies will expose Pillar to either
the producer’s risk or the consumer’s risk:
P(x<11
or x>14)=P(-¥<x<11)+P(14<x<¥)=1-P(11£x£14)
=1-P(-0.71£z£1.43)=1-0.6848=0.3152
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