Statistics 401D
Spring 2016
Laboratory Assignment 3
1. Before computers are assembled, their motherboards go through a special inspection. Historically,
91.5% of the motherboards pass this inspection. In a random sample of 100 motherboards tested
let X be the number of motherboards that will fail this inspection and let the distribution of X be
modeled by a Binomial.
(a) Verify that the conditions for using the Normal approximation to the Binomial are satisfied.
(b) Use the Normal approximation to the Binomial with the continuity correction to find the
probability that the sample contains
i. at least 10 motherboards will fail the inspection.
ii. no more than 5 motherboards will fail the inspection.
iii. exactly 8 motherboards will fail the inspection.
(c) Calculate the exact answer to part (b)iii using the Binomial probability function. Must use
hand calculation with a calculator. (May use Excel or other software for doing the arithmetic
without using the Binomial probability function.)
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2. A firm is considering using the Internet to supplement its traditional brick-and-mortar marketing.
From data obtained from companies engaged in distributing similar merchandise, it is estimated
that one of every 1000 Internet hits result in a sale. Under the assumption that this firm expects
to have 2,500 hits in a single day, answer the following, using a Binomial model:
(a) Write an expression for the probability that there are less than six sales in a day. (Do not
complete the calculations.)
(b) Discuss whether the assumptions necessary to use the above expression to calculate the probability in part (a) are satisfied?
(c) Use a normal approximation to compute the probability that less than six sales are made in a
day.
(d) Use a Poisson approximation to compute the probability that less than six sales are made in
a day.
(e) Use a computer program, internet calculator, Excel, or JMP to compute the exact probability
that less than six sales are made in a day. Compare this result with your calculations in parts
(c) and (d). (Note that the calculator linked to the course webpage uses an approximation).
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3. A particular make of gasoline tank for a compact car is designed to hold 15 gallons. Suppose that
the actual capacity X of a randomly chosen tank of this make is normally distributed with mean
15 and standard deviation .2 gallons.
(a) What is the probability that a randomly selected tank of this make will hold at most 14.8
gallons? Show your work.
(b) What is the probability that a randomly selected tank of this make will hold between 14.7 and
15.4 gallons? Show your work.
(c) If the car on which a randomly chosen tank is mounted gets exactly 25 m.p.g., what is the
probability that the car can travel 370 miles without refueling? Show your work.
(d) At most 1% of gasoline tanks of this make, exceed a certain volume. Calculate this value in
gallons upto an accuracy of two decimals.
(e) If a gasoline tank of this make exceeds the capacity in part d) above, it is considered to be
defective. Using the theory learned in Chapter 4 about the binomial distribution, determine
approximately, how many tanks would you expect to be defective in a lot of 10000 tanks. Show
work.
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4. A quality control inspector is investigating the tension bond strength of batches of mortar made from
a new Portland cement formulation. Suppose that the distribution of strength values Y is assumed
to have a Normal distribution with mean of 17.6 and standard deviation of 0.32 (in kgf /cm2 ).
(a) If a batch is acceptable only if the tension bond strength of the mortar is within 0.5% of the
population mean, what is the probability that a randomly chosen batch is acceptable? Show
your work.
(b) What is the sampling distribution of the mean tension bond strength Y of random samples of
size 100 from the population of batches of mortar? State the mean and standard deviation of
Y clearly.
(c) Compute the probability that the mean tension bond strength of a random sample of 100
batches of the mortar formulation exceeds 17.68.
(d) If the acceptable range of a batch is as determined in part (a) and if the tension bond strength
of each of 10 randomly selected batches is independently determined, what is the probability
that at least 9 out 10 of those batches of mortar are acceptable. Show your work. [Hint: First
define the acceptable number of batches in a sample of size 10 as a Binomial random variable.]
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5. A machine at a General Mill’s plant, designed to fill 16oz. boxes with a cereal, is set-up to produce
boxes that have a population mean µ = 16.2. and standard deviation σ = 1.25 of content in weight.
A random sample of n = 64 boxes from this machine are tested and let Y denote the random
variable that corresponds to the mean of this sample.
(a) Describe the approximate distribution of Y according to the Central Limit Theorem.
(b) Approximately, find P (Y > 16.4) assuming the distribution from part (a) for Y . Show your
work.
(c) Approximately, find P (16.0 < Y < 16.6) assuming the distribution from part (a) for Y . Show
your work.
(d) Find approximately, a constant k such that P (Y > k) = 0.05 assuming the distribution from
(a) for Y . Show your work.
(e) Suppose a sample of size 120 was taken instead. Would the probability calculated in part (b)
increase or decrease? Explain why. (No calculations necessary; but an explanation is needed
even if you do calculate.)
Due Tuesday, February 09, 2016 (turn-in by 10:20 a.m.
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during lab)