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STAT 381 Spring 2017
Final - Review Problems
1. A certain federal agency employs three consulting firms (A, B, and C) with probabilities 0.40, 0.35, and 0.25, respectively. From past experience it is known that
the probability of cost overruns for the firms are 0.05, 0.03, and 0.15, respectively.
(a) What is the probability that the agency experiences a cost overrun?
(b) Suppose a cost overrun is experienced by the agency, what’s the probability
that the consulting firm involved is C?
2. Consider the following joint probability density function of the random variables X
and Y :
(
kxy,
0 < x < 1, 0 < y < 1,
f (x, y) =
0,
elsewhere.
(a) Find k so that f (x, y) is a density function.
(b) Find the marginal density functions of X and Y .
(c) Evaluate P (0 ≤ X ≤ 1/2, 1/4 ≤ Y ≤ 1/2).
(d) Are the two random variables X and Y independent?
3. It’s known that 60% of mice inoculated with a serum are protected from a certain
disease. If 5 mice are inoculated, find the probability that
(a) none contracts the disease.
(b) fewer than 2 contract the disease.
(c) exactly 3 contract the disease.
4. Find the probability that a person flipping a coin gets
(a) the third head on the seventh flip.
(b) the first head on the fourth flip.
5. The television picture tubes of manufacturer A have a mean lifetime of 6.5 years
and a standard deviation of 0.9 years, while those of manufacturer B have a mean
lifetime of 6.0 years and a standard deviation of 0.8 years. Suppose a random sample
of 36 tubes from manufacturer A will have a mean lifetime X̄1 and a random sample
of 49 tubes from manufacturer B will have a mean lifetime X̄2 . These two random
samples are independent.
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(a) Find the sampling distribution of X̄1 − X̄2 .
(b) Evaluate P (X̄1 − X̄2 ≥ 1.0).
6. Two distinct solid fuel propellants, type A and type B, are being considered for
a space program activity. Burning rates of the propellant are crucial. Random
samples of 20 specimens of the two propellants are taken with sample means 20.50
cm/sec for propellant A and 24.50 cm/sec for propellant B. It is generally assumed
that the variability in burning rate is roughly the same for the two propellants
and is given by a population standard deviation of 5 cm/sec. Assume that the
burning rates for each propellant are approximately normal and hence make use
of the Central Limit Theorem. Nothing is known about the two population mean
burning rates, and it is hoped that this experiment might shed some light on them.
(a) If, indeed, µA = µB , what is P (X̄B − X̄A ≥ 4.0)?
(b) Use your answer in (a) to shed some light on the proposition that µA = µB .
7. Traveling between two campuses of a university in a city via shuttle bus takes, on
average, 28 minutes with a standard deviation of 5 minutes. In a given week, a bus
transported passengers 40 times. What’s the probability that the average transport
time was more than 30 minutes?
8. A random sample of 100 automobile owners in the state of Virginia shows that an
automobile is driven on average 23,500 kilometers per year with a standard deviation
of 3900 kilometers. Assume the distribution of measurements to be approximately
normal.
(a) Construct a 95% confidence interval for the average number of kilometers an
automobile is driven annually in Virginia.
(b) What can we assert with 95% confidence about the possible size of our error if
we estimate the average number of kilometers driven by car owners in Virginia
to be 23,500 kilometers per year?
9. The proportion of adults living in a small town who are college graduates is estimated to be p = 0.6. To test this hypothesis, a random sample of 15 adults is
selected. If the number of college graduates in the sample is anywhere from 6 to 12,
we shall not reject the null hypothesis that p = 0.6; otherwise, we shall conclude
that p 6= 0.6.
(a) Evaluate α assuming that p = 0.6. Use the binomial distribution.
(b) Evaluate β and the power of the test for the alternatives p = 0.5.
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(c) Is this a good test procedure?
10. A manufacturer has developed a new fishing line, which the company claims has a
mean breaking strength of 15 kilograms with a standard deviation of 0.5 kilogram.
To test the hypothesis that µ = 15 kilograms against the alternative that µ < 15
kilograms, a random sample of 50 lines will be tested. The critical region is defined
to be x̄ < 14.9.
(a) Find the probability of committing a type I error when H0 is true.
(b) Evaluate β for the alternative µ = 14.8 kilograms.
11. In the American Heart Association journal Hypertension, researchers report that
individuals who practice Transcendental Meditation (TM) lower their blood pressure significantly. A random sample of 225 male TM practitioners yielded a mean
meditation time of 8.5 hours per week with a standard deviation of 2.25 hours.
(a) Test the hypothesis that µ = 8 hours against the alternative hypothesis, µ > 8
hours, at the 0.05 level of significance. Use a P-value in your conclusion.
(b) How large a sample is required if the power of the test is to be 0.8 when the
true mean meditation time exceeds the hypothesized value by 1.5 hours? Use
α = 0.05.
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