Mathematical and Computational Methods for Engineers
E155C, Winter 2004
Review Problems
1. If a driver consistently uses motor oil A, then the
probability of a motor repair in excess of $600 prior to
75,000 miles on the odometer is 0.03; however, for
motor oil B, the probability is 0.08. In a test sample, 40
cars have been consistently lubricated with motor oil A
and 60 with motor oil B.
a) if a test car is selected at random, what is the
probability that it will require $600 of repair work
before being run for 75,000 miles ?
b) given that a car requires motor work in excess of
$600 prior to 75,000 miles, what is the probability it
was lubricated with motor oil A ?
2. If X and Y are independent, show that the characteristic
function for Z  X  Y can be expressed as follows:
 Z (w)   X (w)Y (w)
3. If the average number of claims handled daily by an
insurance company is 5, what proportion of days have
less than 3 claims ? What is the probability that there
will be 4 claims in exactly 3 of the next 5 days ?
4. A contractor purchases a shipment of 100 transistors. It
is his policy to test 10 of these transistors and to keep
the shipment only if at least 9 of the 10 are in working
condition. If the shipment contains 20 defective
transistors, what is the probability it will be kept ?
5. The joint density of X and Y is given by f ( x, y )  2
over the interval 0  x  y , 0  y  1 . Are X and Y
independent ?
1 n
( X i  X )2

n  1 i 1
estimator of the true variance  2
6. Show that s 2 
is an unbiased
7. The lot average yield strength of a certain alloy is
required to be not less than 150 ksi. The standard
deviation based on past experience is 7 ksi and yield
strength is known to possess a normal distribution. Ten
specimens were drawn at random from a large lot and
tested with the following results:
145 142 151 153 138 131 146 153 131 150
a) using a 5% level of significance, should the lot be
accepted ?
b) what is the probability of accepting a lot that has a
lot average yield of 145 ksi ?
c) if standard deviation were not known, find a 95%
confidence interval for the mean strength
8. A manufacturer claims that the resistance of carbon
film resistors has a standard deviation of 0.50 Ohm.
The sample variance of 20 resistors has been found to
be equal to 0.49 Ohm.
a) would you reject the manufacturer’s claim at the 5%
level of significance ?
b) what is the probability of rejecting the
manufacturer’s claim when  1  1.5 0 ?
9. The data below show measurements of the corrosion
effects for steel pipe with two kinds of coating:
X 72 63 53 47 38 56 55 75 68 57 61 52 67
Y 69 69 58 52 33 57 54 66 72 71 69 53 60
a) if X and Y are both normally distributed with known
common variance  2  9 , test the hypothesis at the
5% significance level that the kind of coating used
has no effect on corrosion
b) redo part a) if X and Y are both normally distributed,
and  2 is known to be the same for both X and Y
but its value is unknown
c) redo part a) if X and Y are not normally distributed,
i.e. use a non-parametric test
10. The following data gives the frequency distribution of
demand for a certain product per day based on the
records of 1000 days:
Demand/day 0
1
2
Freq.
626 274 80
3
15
4
4
5
1
Fit a Poisson distribution to the above data and test for
the goodness of fit at the 5% level of significance.
11. The following data represent the effect of annealing
temperature x on the ductility of brass, measured in
terms of elongation y:
x, deg C
y, (%)
300 400 500 600 700 800
40 50 55 60 67 70
a) determine the regression line of y on x
b) estimate y when x  550
c) find a 95% confidence interval for the slope of the
regression line
12. Suppose that 20 specimens of metal were tested to
ascertain whether there is a relationship between yield
and tensile strength. The estimate of the correlation
coefficient  was found to be 0.85.
a) is there evidence of a statistically significant
correlation between yield and tensile strength ?
b) find a 95% confidence interval for  using Fisher’s
transformation