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Unformatted text preview: UNIVERSITY OF CALIFORNIA, LOS ANGELES
Civil and Environmental Engineering Department
CEE 110 Introduction to Probability and Statistics for Engineers
Spring Quarter 2011
MW 122 PM
Franz 1178 Prof. K. D. Stolzenbach
5732J Boelter Hall, 2067624
stolzenb@ucla.edu Problem Set 5
(Due Wednesday May 11, 2011)
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1. (problem 8.14 in text)
The life in hours of a 75watt light bulb is known to be normally distributed with σ = 25
hours. A random sample of 20 bulbs has a mean life of x = 1014 hours.
(a) Construct a 95% twosided confidence interval on the mean life.
(b) Construct a 95% lowerconfidence bound on the mean life. Compare the lower bound of
this confidence interval with the one in part (a).
[1003, 1025; 1005]
2. (problem 8.16 in text)
Suppose that in Problem 1 we wanted the error in estimating the mean life from the twosided confidence interval to be five hours at 95% confidence. What sample size should be
used? [97] 3. (problem 8.28 in text)
An Izod impact test was performed on 20 specimens of PVC pipe. The sample mean is x =
1.25 and the sample standard deviation is s = 0.25. Find a 99% lower confidence bound on
Izod impact strength. [1.108] 4. (problem 8.32 in text)
An article in the Journal of Composite Materials (December 1989, Vol. 23, p. 1200)
describes the effect of delamination on the natural frequency of beams made from composite
laminates. Five such delaminated beams were subjected to loads, and the resulting
frequencies were as follows (in hertz):
230.66, 233.05, 232.58, 229.48, 232.58
Check the assumption of normality in the population. Calculate a 90% twosided confidence
interval on mean natural frequency. [230.2, 233.1] 5. (problem 8.32 in text)
Of 1000 randomly selected cases of lung cancer, 823 resulted in death within 10 years.
(a) Calculate a 95% twosided confidence interval on the death rate from lung cancer.
(b) Using the point estimate of p obtained from the preliminary sample, what sample size is
needed to be 95% confident that the error in estimating the true value of p is less than
0.03?
(c) How large must the sample be if we wish to be at least 95% confident that the error in
estimating p is less than 0.03, regardless of the true value of p?
[0.799, .847; 622; 1068] 6. (problem 858 in text)
A random sample of 50 suspension helmets used by motorcycle riders and automobile racecar drivers was subjected to an impact test, and on 18 of these helmets some damage was
observed.
(a) Find a 95% twosided confidence interval on the true proportion of helmets of this type
that would show damage from this test.
(b) Using the point estimate of p obtained from the preliminary sample of 50 helmets, how
many helmets must be tested to be 95% confident that the error in estimating the true value
of p is less than 0.02?
(c) How large must the sample be if we wish to be at least 95% confident that the error in
estimating p is less than 0.02, regardless of the true value of p?
[0.227, 0.493; 2213; 2401] 7. In this exercise you will use Matlab to sample a population with a known probability
(proportion) of “success” p and apply a normal distribution model to the estimate of this
probability . After opening Matlab, create the following Mfile (see Problem Set 3):
p=0.5; % this sets the probability (proportion)
n=100; % this sets the sample size
m=1000; % this sets the number of times you will sample
sig=(p*(1p)/n)^0.5; % this computes the SD of a sample
y=binornd(n,p,m,1)/n; % this obtains m estimates of p
x=p3*sig:sig/10:p+3*sig; % this sets bins for the histogram of estimates
hist(y,x) % this makes a histogram of the estimates
Save this M file to your My Documents file and run the Mfile,
creating a figure with the histogram (see example). Annotate this
figure and hand it in with your problem solutions. Also, use the
figure to check if the number of samples with > 0.6 agrees with
the predictions of a normal model for this sample. ...
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This note was uploaded on 10/22/2011 for the course ENG 101 taught by Professor Yukov during the Spring '11 term at UCLA.
 Spring '11
 Yukov
 Environmental Engineering

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