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STAtistics / Mr. Hansen |
Name:
_________________________ |
Test #4
Probability, random variables, Type I/Type II error,
PPV, and simulations
Time limit: 50 minutes.
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Useful Formulae: |
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P(A È B) = P(A)
+ P(B) – P(A Ç B) |
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E(X) = mX = S xi pi |
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1. |
At the NIA, 30% of the
employees are moles (i.e., double agents). To root them out, we try
administering polygraph tests. The polygraph flags (“+” reading) 80% of
moles, and the polygraph clears 75% of non-moles. There are no ambiguous
results. |
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(a) |
State H0. |
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(b) |
Compute a, the probability of a Type I error. |
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(c) |
Compute b, the probability of a Type II error. |
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(d) |
Compute P(mole | person is flagged by the
polygraph test). |
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(e) |
What name do we give to
(d)? (Abbrev. is acceptable.) |
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(f) |
Compute P(not mole | person is cleared by the
polygraph test). |
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(g) |
In an SRS of 10 employees,
compute the probability that more than 4 are moles. Work is not required. |
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(h) |
What assumption (a rule of
thumb that we learned) is needed in order to proceed with the computation in
part (g)? Is this assumption probably reasonable if NIA is a large agency? |
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(i) |
In an SRS of 10 employees,
compute the probability that at least 1 is a mole. This time, work is required. |
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2. |
Mr. Hansen is a terrible
free-throw shooter, hitting only 15% of his attempts. This probability is
unchanging and independent of trials. |
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(a) |
What model is appropriate
for determining the probability of hitting the first success in 1, 2, 3, 4, .
. . tries? Explain in detail,
listing all the requirements that are satisfied. |
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(b) |
Compute the expected number
of shots needed for Mr. Hansen’s first success. |
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(c) |
Estimate the standard
deviation of the number of shots needed for Mr. Hansen’s first success. Consider
only the cases 1 through 8. Show work. [The formula |
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3. |
In a dice game with
standard dice numbered 1-6, I will roll 2 dice. If I obtain 2 even numbers, I
win $10 from you. If I obtain 2 odd numbers, I will pay you $15. If the dice
are a mixture of odd and even, nobody wins. Write out the instructions for a
simulation to see what the probability that I will be ahead money after 4
rolls is. Note: A game consists of 4 rolls of the pair of dice.
Then use your calculator’s pseudorandom digit generator to estimate this
probability by simulating 8 games. Show your results in a table. |
