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ENGR 315 MIDTERM #2 – PRACTICE TEST QUESTIONS
For Midterm #2 Date: See Syllabus
For the Answers: See Solutions
Instructions for the Test: This test will be closed book. However, you will be
allowed to use up to 6 pages of your own notes. You will have 50 minutes. For each
problem, you should show how you got your answer. The method of the solution will count
more than the numerical values. We encourage you to draw diagrams and to state, explicitly
and precisely, the definition of your random variables. Good luck!
Practice Test Notes: Below are several practice test questions. These questions
should provide you with an understanding of the types of concepts that may be on the test.
The test will consist of four questions.
 | The following questions relate to the failure of television sets and their warranty's.
(a). The television picture tubes of Manufacturer A have a mean lifetime of 5 years
(mA=5) and a standard deviation of 1.11 years (sA=1.11), and follow a normal distribution. If
Manufacturer A gives a warranty for 4 years, what is the probability that a picture tube
will fail during the warranty period?
(b). The television picture tubes of Manfacturer B also have a mean lifetime of 5
years (mB=5) but they follow a exponential
distribution. If manufacturer B gives a warranty for 4 years, what is the
probabiliity that a picture tube will fail during the warranty period? Hint: In case
you need help with integrals, òaeaxdx = eax
and òae-axdx = -e-ax.
(c). Would you buy picture tubes from Manufacturer A or B (given they cost the same
price)? Explain why.
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| The following questions concern the strength of glass:
(a). The fracture strengths of a certain type of glass have a mean of 14 (in
thousands of pounds per square inch) and have a standard deviation of 2. Using the
Central Limit Theorem, find the probability that the average (sample mean) fracture
strength for 100 pieces of this glass exceeds 14.5.
(b). As in part a above, how many samples of glass should be tested so that the
sample mean would be within 0.2 of the population mean (m=14)
with probability 0.95?
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| The manufacturer of an indutrial plant is planning to buy a new machine of either type A
or type B. For each day's operation, the number of repairs that machine type A
requires, YA, is a Poisson random variable with mean 1.0, and the number of repairs that
machine type B requires, YB is a Poisson random variable with mean 1.2.
(a). Find the probability that no repairs are needed in a day's operation for
machine A. Find the probability that no repairs are needed in a day's operation for
machine B.
(b). Suppose that the daily cost of operating machine A is CostA = 100 +
30 (YA)2 and the daily cost of operating machine B is CostB=70+30(YB)2.
Assuming that the machines are to be cleaned each night so they operate like new
machine at the start of each day, find the expected daily cost for both machine types.
Hint: Use Var(Y) = E[Y2] - m2.
(c). If you were a manager, would you buy machine A or B? Explain why.
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| In order to test gasoline mileage performance for a new version of one of its compact
cars, an automobile manufacturer selected 6 nonprofessional drivers to drive a car from
Phoenix to Los Angeles. The miles per gallon values for the six cars at the
conclusion of the trip were 27.2, 29.3, 31.5, 28.7, 30.2, and 29.6. The manufacturer
wishes to advertise that cars of this type average (at least) 30 mpg on such long trips.
Given the sample data, would you support or contradict this claim?
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| The manufacturer's recommended air pressure for a certain type of tire is 32 psi.
A random sample of n=35 such tires from different cars operated by a large rental company
yields a sample average pressure and sample standard deviation of 31.4 and 1.2
respectively. Does this data suggest the average pressure for all such tires on the
company's cars differs from the recommended value?
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| A manufacturer claims that the average lifetime of a certain camera is 12 years.
The manufacturer wants to offer a guarantee, but is undecided whetehr to offer a 7, 8, or
9 year guarantee. The manufacturer is not willing to replace more than 2.5% of all
cameras.
(a) Assume that the camera's lifetime has a normal distribution with mean = 12 years
and standard deviation = 1.5 years. Find the value of b, such that the 2.5 % of all
cameras have a lifetime less than b years.
(b). Using part a, how many years would you recommend for the guarantee?
(c). To be convinced that this is a reasonable guarantee, the manufacturer
formulates a one-tailed hypothesis test, with H0: m
= 12 years and H1: m < 12 years. Testing 36
cameras gives a sample mean of 11.5 years and a sample standard deviation of 2.7 years.
Find the critical region associated with a 0.05 level of significance.
(d). Summarize the conclusion from the test. Using part c, would you change
your recommendation for the guarantee? |
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