Chapter 5 Assignment 3 - Hypergeometric Distribution-1-1
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Dec 6, 2023
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The following problems use the Hypergeometric Distribution. Complete each using the formulas posted
or the Hypergeometric Template. Clearly identify your answers.
The Hypergeometric Distribution (The Hypergeometric Model)
Suppose that a population consists of
N
items and that
r
of these items are
successes
and (
N
−
r
) of
these items are
failures
. If we randomly select
n
of the
N
items
without replacement
, it can be shown
that the probability that
x
of the
n
randomly selected items will be successes is given by the
hypergeometric probability formula
The Mean and Variance of a Hypergeometric Random Variable
Suppose that
x
is a hypergeometric random variable. Then
For each problem post the following prior ro answering the questions for the problem:
The size of the sample - n
The number of successes in the population - r
The population size - N
Problem 1:
Among 20 metal parts produced in a machine shop, 6 are defective. If a random sample of
three of these metal parts is selected, find:
1.
The probability that this sample will contain at least two defects.
0.0456
2.
The probability that this sample will contain at most one defect.
0.9544
Problem 2:
Suppose that you purchase (randomly select) 3 TV sets from a production run of 10 TV sets.
Of the 10 TV sets, 8 are destined to last at least five years without needing a single repair.
1.
What is the probability that all three of your TV sets will last at least five years without needing a
single repair? 0.9333
Problem 3:
Suppose that you own a car dealership and purchase (randomly select) 10 cars of a certain
make from a production run of 200 cars. Of the 200 cars, 160 are destined to last at least five years
without needing a major repair.
1.
Using the hypergeometric distribution what is the probability that at least 6 of your 10 cars will
last at least five years without needing a major repair. 0.9709
2.
Then, using the binomial tables or results from the template approximate this probability by
using the binomial distribution. Hint:
p
=
r
/
N
= 160/200 = .8. 0.8846
3.
What conclusions can you make comparing the results – Are they similar and could you use the
Binomial Probability to approximate the Hypergeometric probability? binomial probability can
be effectively used to approximate the hypergeometric probability in situations like the one
described, making calculations more manageable while still providing reasonably accurate
results.
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