STATISTICS 401B
Fall 2014
Laboratory Assignment 3
1. A company has determined that an average of 1% of an expensive manufactured item will be
damaged during shipment. Model the number of damaged items in a shipment consisting of 1000
of these items as a Binomial random variable.
(a) Verify that the conditions for using the Normal approximation to the Binomial are satisfied in
this situation. Is the continuity correction needed?
(b) Use the Normal approximation to the Binomial (with the continuity correction) to find the
approximate probability that during this shipment
i. at least 20 items will be damaged.
ii. no more than 5 items will be damaged.
iii. exactly 10 items will be damaged.
(c) Calculate the exact answer to part (b)iii using the Binomial probability function.
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(d) Verify whether conditions for using a Poisson approximation to the above Binomial are satisfied. If so compute the probability that no more than 5 items will be damaged using a Poisson
approximation.
(e) Use a computer program, internet calculator, Excel, or JMP to compute the exact probability
that no more than 5 items will be damaged, upto six significant digits accuracy. Compare
this result with your calculations in parts (b) and (d) (i.e., which appears to be the better
approximation).
2. Cars arrive at a toll booth at a rate of six per 10 seconds during rush hours. Assume that X, the
number of cars arriving during any 10-second period, has a Poisson distribution.
(a) no cars arrive in a 10-second period.
(b) no more than one car arrives in a 10-second period.
(c) at least two cars arrive in a 10-second period.
(d) at least four cars arrive in a 20-second period.
Note: You must show the work of your calculations using the Poisson pmf. Your answers may be
checked using Table 15 of the text book or using a Poisson calculator available on the internet.
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3. A particular brand of cement is sold in 50 lb bags. Suppose that the actual net weight X of a
randomly chosen bag of cement of this brand is Normally distributed with mean µ = 50 lbs. and
standard deviation σ = 1 lb.
(a) A bag is considered underfilled if it weighs less than 49.0 lbs. What is the probability that a
randomly selected bag is underfilled? Show your work.
(b) A bag will meet the standard established by a consumer organization if it weighs between 49.75
and 50.75 lbs. What is the probability that a randomly selected bag will meet this standard?
Show your work.
(c) The manufacturer is not happy that as much as 1% of the cement bags of this brand are
heavier than a certain weight. Approximately, determine this weight? Show your work.
(d) The cement company wants to adjust the settings of the filling machine that fills cement bags
of this brand so that the probability that a randomly selected bag is underfilled is at most
0.1. If the adjustments let σ remain constant at 1 lb, to what value must the mean setting be
increased to achieve this goal?
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4. A quality control inspector is investigating the potency of batches of a new antibiotic. Suppose
that the distribution of potency values Y is assumed to have a Normal distribution with mean of
9.6 and standard deviation of 0.24 (assuming that the potency is measured on a continuous scale).
(a) If a batch is acceptable only if its potency 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 potency Y of random samples of size 100 from
the population of batches of antibiotic? State the mean and standard deviation of Y clearly.
(c) Compute the probability that the mean potency of a random sample of 100 batches of the
antibiotic exceeds 9.64.
(d) If the acceptable range of a batch is as determined in part (a) and the potency of each of 10
randomly selected batches is independently determined, what is the probability that at least 9
out 10 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. Assume that the breaking strengths of 1-foot square samples of a synthetic fiber are distributed
with mean µ = 2250 pounds per square inch (psi) and standard deviation σ = 21.6 psi. Suppose
that a random sample of n = 36 of 1-foot square samples from this fabric are tested.
(a) Describe the approximate distribution of Y according to the Central Limit Theorem, specifying its mean and variance.
(b) Find P (Y > 2260) assuming the distribution from part (a) for Y . Show your work.
(c) Find P (Y < 2244) assuming the distribution from (a) for Y . Show your work.
(d) Find k such that P (Y > k) = 0.05 assuming the distribution from (a) for Y . Show your work.
(e) Suppose a sample of size 72 was taken instead. Would the probability calculated in part (b)
increase or decrease? Explain why. (Do not do any calculations; instead explain using the
dependence of this probability on the distribution of Y )).
Due Thursday, Sept 25, 2014 (turn-in before 2:20 p.m.)
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