Mathematical and Computational Methods for Engineers
E155C, Winter 2004
Handout #3
Estimation, Confidence Intervals, Hypothesis Testing, Nonparametric Tests
Confidence Intervals
1. A manufacturer measures the inside diameter of 20 pipe
fittings by random sampling. The empirical mean is
x  1.5 in and the standard deviation is   0.01 in.
a) Determine the 95% confidence interval for the
mean
b) How large a sample is required to ensure with
99% confidence that the mean is contained
within 0.005 in of the sample mean ?
2. Consider a normally distributed random variable with
variance  2 .
a) For a sample size of 30, what is the probability
that the sample variance will fall within 45% of
the true variance
b) With s 2  0.01 construct the 95% confidence
interval for  2
3. Six samples of hardened concrete have been tested for
compressive strength. The following values were
recorded:
4052
4120
4095
4188
3978
4090
The strength is believed to be normally distributed.
a) Determine the 90% confidence interval for the
mean
b) If  was assumed to be known such that
  69.96 , find the 90% confidence interval for
the mean
Estimation
4. Suppose X is known to be normally distributed with
known variance  2 . Find the maximum likelihood
estimator for the mean.
5. Suppose X is normally distributed with known mean 
but unknown variance. Find the maximum likelihood
estimate for the variance.
Hypothesis Testing
6. A structural engineer is experimenting with a new alloy
that may increase the yield of a high-strength structural
steel. The objective is to demonstrate that the new alloy
is superior to the old one. From experience it is known
that for this class of alloys the yield strength is normally
distributed with the standard deviation   0.75 ksi and
the mean strength of   46 ksi. The plan is to take 9
rods of the new alloy and demonstrate that the mean
yield is larger than 46 ksi at the 5% significance level.
Assume that x  46.9 ksi.
7. In designing a service system, it is beneficial to have a
hypothesis regarding the number of units to be serviced
in a given time frame. Too low an estimate will result in
inadequate service, and too high an estimate will result
in a waste of resources. Suppose that company’s
allocation has been based on 110 users. It is desired to
check if there has been a relevant deviation at the
significance level of 5%. Assume that   8.4 users
and n  25 .
8. The environmental agency of a certain city claims that
the amount of hydrocarbons in the atmosphere has been
reduced to a level of 1.8 ppm. To verify the claim, a
sample of 12 observations is to be taken. It is known
that the measurements tend to be normally distributed.
Suppose that the sample mean is x  1.68 ppm with
standard deviation of s  0.17 ppm. Test the hypothesis
at 1% significance level.
9. A manufacturer suspects that substandard parts have
been substituted for the ones ordered. From experience,
it is known that only 10% of parts are generally
defective. Suppose that 60 parts are chosen randomly
from the shipment and 9 were found to be defective.
Test the null hypothesis at the 5% significance level.
10. In milling, surfaces are machined by a rotating cutter.
Suppose a cylinder head is to be cut to a high degree of
precision. After a period of use it is desired to check the
variation occurring in a particular machine. The
machine must receive maintenance when  2 exceeds
9  10 6 in2. A sample of 30 yields s 2  1.4  105 in2.
Due to costly service and down time, assume   0.01.
At this level of significance, should the maintenance be
performed ?
11. A claim is made that the service life of brake pads
produced by manufacturer A is over 4,000 mi longer
than the life of pads produced by manufacturer B. 40
randomly chosen samples were selected from each of
the two brands. Test the claim at a 5% significance
level.
x  42,500
y  36,400
 x  4,400
 y  5,300
Nonparametric Tests
12. A chemist wishes to demonstrate that the melting point
of a new compound exceeds 184.5 degrees C. The
underlying
distribution
of
the
temperature
measurements is not known. Twelve measurements are
taken and the results are shown below. Test the
hypothesis at the 10% confidence level.
185, 186, 185, 182, 183, 186, 186, 185, 183, 184, 185, 187
13. Suppose that eight operators were tested, first in rooms
without A/C and then in rooms with A/C over the same
period of time. The number of errors made was
recorded.
w/o A/C
w A/C
5 5 2 4 3 5 3 5
1 3 3 3 0 2 4 4
Test the hypothesis that the median has not changed.
14. The temperature is measured during a given time period
in 3 consecutive years at a specified location:
85 90 87
Determine whether or not there has been a positive
trend at the 5% confidence level.