Stat 330 (Spring 2015): Homework 11
Due: April 24, 2015
Show all of your work, and please staple your assignment if you use more than one sheet. Write your name,
the course number and the section on every sheet. Show all work to earn partial credit. Problems marked
with * will be graded and one additional randomly chosen problem will be graded.
1. A sample of 3 observations of waiting time to access an internet server is x1 = 0.4, x2 = 0.7, x3 = 0.9
seconds. It is believed that the waiting time has the continuous distribution
(
θtθ−1 , 0 < t < 1
f (t) =
0,
otherwise
(a) Find the maximum likelihood estimate of θ.
2. (Baron’s book) 9.1 (Find MLE only and omit part (c) in this homework set)
3. * (Baron’s book) 9.3 (only find maximum likelihood estimator in each case)
4. * There is concern about the speed of automobiles traveling over a particular stretch of highway. For a
random sample of thirty automobiles, radar indicated the following speeds, in miles per hour:
82 88 64 78 90 57 74 70 81 60 75 78 85 77 78 65 73 79 73 66 71 70 61 77 66
69 72 67 64 74
Let the mean speed of all automobiles traveling over this stretch of highway be µ mph.
(a) Find the sample mean and variance, x̄ and s2 .
(b) Find a 95 % confidence interval for the mean speed of all automobiles traveling over this stretch of
highway.
(c) Test the hypothesis that people are speeding, if the legal speed on this highway is 65 mph. That is,
test H0 : µ = 65 vs. Ha : µ > 65
5. A manager evaluates effectiveness of a major hardware upgrade by running a certain process 50 times
before the upgrade and 50 times after it. Based on these data, the average running time is 8.5 minutes
before the upgrade, 6.2 minutes after it. Historically, the standard deviation σ of the running times of
this process has been 1.8 minutes and presumably it has not changed.
(a) Construct a 90% confidence interval for the difference in the mean running times µBefore − µAfter .
(b) Using this interval, can you conclude that the upgrade was effective? Why?
(c) Test the hypothesis that the hardware upgrade improved the average running time of this process?
6. Given two samples with n1 = 30, n2 = 39, x̄1 = 4.2, x̄2 = 3.4, s21 = 49, and s22 = 32, where x̄1 is the
mean of a sample of size n1 from the first population (mean µ1 ) and has sample variance s21 , and x̄2 is
the mean of a sample of size n2 from the second population (mean µ2 ) with sample variance s22 .
(a) Find a 95% confidence interval for µ1 .
(b) Find a 90% confidence interval for µ1 − µ2 .
(c) Suppose n1 is 10 instead of 30. Re-establish the confidence interval in part (a) by assuming the first
population follows a normal distribution.
7. (Baron’s book): 9.7
1