IEOR 165 Homework 1
Due February 19, 2015 in Class
Electronic copies will not be accepted except under extenuating circumstances. Each student must turn in their own homework solutions.
Question 1. Only two factories manufacture iPhones. One percent of the iPhones from
factory I and five percent of the iPhones from factory II are defective. Factory I produces
twice as many iPhones as factory II.
a. What is the probability that a randomly chosen iPhone is satisfactory?
b. What is the probability that a randomly chosen iPhone is from factory I, if it is found
to be defective?
Question 2. In one chemical process, it is very important that a particular solution used
as a reactant have a pH of exactly 8.15. A method for determining pH available for solutions
of this type is known to give measurements that are normally distributed with a mean equal
to the actual pH and a standard deviation of 0.02. We wish to design a test such that if the
pH is actually equal to 8.15 then this conclusion will be reached with probability equal to
0.90. Suppose the sample size is 5.
a. What is the test procedure that should be used?
b. If X = 8.26, what is the conclusion?
Question 3. An advertisement for a new toothpaste claims that it reduces cavities of children in their cavity-prone years. Cavities per year for this age group are normal with mean
3 and standard deviation 1. A study of 2,500 children who used this toothpaste found an
average of 2.95 cavities per child. Assume that the standard deviation of the number of
cavities of a child using this new toothpaste remains equal to 1.
a. Is this data strong enough, at the 5 percent level of significance, to establish the claim
of the toothpaste advertisement?
b. Does the data convince you to switch to this new toothpaste?
Question 4. Define s′2 =
1
n
n
P
(Xi − X)2 .
i=1
1
a. Show that
n−1 2
σ
E[s ] =
n
′2
b. Show that
E
Hint:
n
P
i=1
V ar(Xi )
(Xi − X)2 =
n
P
"
1
n−1
n
X
#
(Xi − X)2 = σ 2
i=1
(Xi − µ) − (X − µ)
i=1
2
where µ = E[Xi ] = E[X], σ 2 =
Question 5. State the null hypothesis to be used in the following tests. What type of
extreme deviation would you use for the test?
1. A manufacturer of a certain brand of rice cereal claims that the average saturated fat
content is 1.5 grams.
2. A real estate agent claims that 60% of all private residences being built today are 3bedroom homes. To test this claim, a large sample of new residences is inspected: the
proportion of these homes with 3 bedrooms is recorded and used as our test statistic.
3. A dry cleaning establishment claims that anew spot remover will remove more than
70% of the spots to which it is applied.
Question 6. The independent measurements given below are normally distributed with
unknown variance.
a. Compute a 95 percent confidence interval using the appropriate equations.
b. Construct a 95 percent confidence interval using the percentile bootstrap.
Return your results and code. (Matlab, R...)
38.65 40.05 38.98 42.29 44.34 45.34 51.06 48.62 48.73 36.29 44.3 37.01 42.73 43.34 38.72 35.68
44.08 41.07 39.29 42.49 51.73 43.86 38.95 34.17 38.32 29.72 47.55 38.39 46.42 43.6 47.28 39.32
39.95 38.13 33.79 45.53 40.26 50.72 37.81 45
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