Download Math 333 - Chapter Seven Sample Exam 1. The cereal boxes

Math 333 - Chapter Seven Sample Exam
1.
The cereal boxes coming off an assembly line are each
supposed to contain 12 ounces. It is reasonable to assume
that the amount of cereal per box has an approximate
normal distribution. A random sample of 16 boxes yields a
mean of 11.78 ounces and a standard deviation of 0.22
ounces.
(a) Use the given data to find a 95% confidence interval
for the mean amount of cereal per box.
(b) Based on the results in part (a), should we conclude
that the assembly line equipment needs to be adjusted,
and if so, how?
(c) What is the smallest sample size that could be used to
estimate the mean amount of cereal per box with a
95% confidence interval so that the maximum error of
the estimate is 0.03 ounces?
(d) Find a 95% confidence interval for the standard
deviation of the amount of cereal per box on the
assembly line.
2.
Data is taken to compare the proportion of complaints at
two airline terminals. In a random sample of 500
customers at the Transburg terminal, 51 have complaints.
In a random sample of 400 customers at the Central City
terminal, 48 have complaints.
(a) Use the given data to find a 90% confidence interval
for the difference in proportion of complaints for the
two terminals.
(b) Based on the results in part (a), should we conclude
that the proportion of complaints is higher at one
terminal than the other, and if so, which terminal has
the higher proportion of complaints?
3.
Suppose X1, X2, ..., Xn is a random sample from a
distribution having p.d.f.
f(x) = (1 ! 1) / x1 if x > 1,
where 1 < 1 .
(a) Find the maximum likelihood estimator for 1 (and
don't forget to use the second derivative test to show
that the likelihood function is maximized).
(b) Find the method of moments estimator for 1 .
4.
Data on rainfall is taken to compare a new cloud seeding
technique and a standard cloud seeding technique. We are
willing to assume that rainfall has an approximate normal
distribution with the same variance for each technique.
The mean and standard deviation for a random sample of 8
cloud seedings with the new technique are respectively 3.9
inches and 0.91 inches. The mean and standard deviation
for a random sample of 12 cloud seedings with the standard
technique are respectively 2.1 inches and 0.75 inches.
(a) Use the given data to find a 95% confidence interval
for the difference in mean rainfall for the two cloud
seeding techniques.
(b) Based on the results in part (a), should we conclude
that one technique is better than the other, and if so,
which one is better?
(c) Find a 95% confidence interval for the ratio of the
variances of rainfall from the two techniques.
5.
In a random sample of 1000 voters, 650 say that they will
vote for Ms. Jones if she is a candidate in an upcoming
election.
(a) Use the given data to find a 99% confidence interval
for the true proportion of voters who will vote for Ms.
Jones.
(b) Based on the results in part (a), should Ms. Jones
conclude she will win the election?
(c) What is the smallest sample size that could be used to
estimate the true proportion of voters who will vote
for Ms. Jones with a 99% confidence interval so that
the maximum error of the estimate is 0.01?
6.
Suppose X1, X2, ..., Xn is a random sample from a
distribution having p.d.f.
f(x) = e!x/(2$) / (2$)
if x > 0 .
(a) Prove that for each i, the moment generating function
M(t) of Xi / $ is the same as the moment generating
function for a chi-square random variable (if t < 1/2),
and state what the degrees of freedom is.
(b) Given the result in part (a), explain why it must be
true that if Y = X1 + X2 + ... + Xn , then Y / $ has a
chi-square distribution, and state what the degrees of
freedom is.
(c) Suppose n = 6, and the following sample values are
observed:
32.542 , 3.399 , 20.505 , 29.861 , 3.726 , 2.632
Use the result in part (b) to derive a 95% confidence
interval (based on Y) for the unknown parameter $.
(d) What must be done to shorten the length of the 95%
confidence interval in part (c) without lowering the
confidence level?
7.
Suppose X is a random variable having p.m.f.
f(x) =
if x = !1, 0, 1.
(a) Show that E(X) = 0.
(b) Find E(X2), and state why X2 is not an unbiased
estimator of 2.
(c) Write a function of the statistic X2 which is an
unbiased estimator of 2.
8.
Suppose X1, X2, ..., Xn is a random sample from a
Poisson(8) distribution.
(a) Show that each of X) and S2 is an unbiased estimator of
8.
(b) Explain how you would decide which of the two
estimators in part (a) is better. (Note: you do not have
to do anything except just explain.)
Here are some additional practice exercises from the textbook
(with answers in the back of the textbook):
7.1-1, 7.1-5(b), 7.1-7, 7.1-9, 7.1-15, 7.1-16,
7.2-7,
7.3-9, 7.3-13,
7.4-1, 7.4-12,
7.5-7, 7.5-13,
7.6-1, 7.6-11,
7.7-5,
7.8-13,
7.9-3