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PROBABILITY AND STATISTICS, A13
FINAL EXAMINATION
(1) (2.5 marks) A random sample of 150 suspension helmets used by
motorcycle riders was subjected to impact test and for 28 of these
helmets damage was observed.
a) Construct a 98% confidence interval for the true proportion of
helmets of this type that would show damage from this test.
b) How large a sample you would have to study to estimate this
proportion with maximum error of 0.01 and 98% confidence?
(2) (3.5 marks) An article in Austr. J. of Agric. Res. (1993) determined
that the Lysyne level (essential amino acid) in a sample of eight
soybean meals is as follows (g/kg):
22.2 24.7 20.9 26.0 27.0 24.8 26.6 23.8
a) Construct a 95% confidence interval for the average Lysine level.
b) Construct a 95% confidence interval for the population standard
deviation. Assume the population is normal.
(3) (2.5 marks) Toxaphene is an insecticide that has been identified as
a polutant in the Great Lakes. An article in J. of Envi. Sci. Health
(1998) studied the effects of toxaphene on anymals and reported
weight gains (in g) for rats given a low dose (4ppm) and for control rats who were not exposed to toxaphene. The standard sample
deviation for 23 control rats was 32g, while for 16 low-dose rats it
was 54g. Does this suggest that there is more variability in low-dose
weight gains than in control weight gains. Carry out a test of hypothesis at significance level 0.05.
(4) (3.5 marks) In going to school, a student must first get on a bus
near her house and then transfer to a second bus. The probability
density function for the total waiting time X (in 10’s of min) for the
two buses is

x if 0 < x < 1

2 − x if 1 < x < 2
f (x) =

0 otherwise
a) Compute the mean and the standard deviation of X.
b) What is the probability that in 75 days (a semester) the student
would have spent more than 12 hours waiting for the bus when going
to school?
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PROBABILITY AND STATISTICS, A13 FINAL EXAMINATION
(5) (2 marks) The information on the side of a 300g box of a particular brand of cornflakes claims that the sodium content is 130mg
per box. Working for a government agency, you sample twenty such
boxes and find a sample average sodium content of 130.88mg with
sample standard deviation of 1.12mg. Does you data contradict the
claim on the box? Use an appropriate hypothesis test assuming the
population distribution of sodium content is normal.
(6) (3.5 marks) An article in Med. and Sci. in Sports and Exercise.
(2005) studied cycling performance before and after eight weeks of
strength training. 70 previously untrained males had on average 315
watts of peak power at the end of training with sample standard
deviation of 35 watts.
a) Is there evidence that the peak power at the end of training will
exceed 300 watts in a population of males undergoing 8 weeks of
training? Test H0 : µ = 300 versus H1 : µ > 300 at α = 0.05.
b) Compute the power of the test if the true peak power is 314 watts.
c) What sample size would be required to detect a true peak power
of 314 watts if we wanted the power of the test to be at least 0.99?
(7) (2.5 marks) The melting points of two alloys used in formulating solder were investigated by melting 21 samples of each material. The
sample mean and standard deviation for alloy 1 was 422F ◦ and 4F ◦ ,
while for alloy 2 they were 426F ◦ and 2.5F ◦ . Do the sample data
support the claim that both alloys have the same melting point? Use
α = 0.05 and assume that both populations are normally distributed.
(8) (2 marks) An article in Fire Techn. investigated two different foam
expanding agents. A random sample of five observation with an
aqueous film-forming foam (AFFF) had a standard deviation of 0.6.
A random sample of 7 observations with alcohol-type concentrates
(ATC) had a sample standard deviation of 0.8. Construct a 95%
confidence interval on σ12 /σ22 .
(9) (3 marks) Let X denote the number of flaws observed on a large
coil of galvanized steel. 80 coils are inspected and the following data
were observed for the values of X:
x 0 1 2 3
freq. 16 26 22 16
Is a Poisson distribution with an appropriate model for this data?
Estimate λ from the data and then perform a goodness-of-fit test
with α = 0.05. What is the p-value for this test?
PROBABILITY AND STATISTICS, A13
FINAL EXAMINATION
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(10) (2 marks) An article in Fortune (Sep. 1992) claimed that nearly
one-half of all engineers continue with academic studies beyond the
B.S. degree, ultimately receiving an M.S. or a Ph.D. degree. Data
from Engineering Horizons indicated that 119 of 484 new engineering
graduates continued with graduate studies. Are this data consistent
with the claim in Forune? Test H0 : p = 0.50 against H1 : p < 0.50
at α = 0.05 level of significance. What is the p-value? Is the claim
in Fortune correct?
(11) (2.5 marks) A multiple choice test has 25 questions, each with 4 answers of which only one is correct.
a) What is the probability that a subject who is randomly guessing
would answer 10 of the questions correctly?
b) What is the probability that in a group of 50 randomly guessing
subjects the average number of correctly answered questions will be
less than 6?
(12) (3 marks) An article in J. of Env. Eng. (1989) reports on a study on
the occurence of chloride in surface streams in Rhode Island. Consider the accompanying data on x = percentage of the watershed
covered by roadways and y = chloride concentration (in mg/L)
x 0.19 0.15 0.57 0.63 0.47 0.70 0.60 0.81
y 4.4 6.6 9.7 10.9 11.0 12.1 13.3 15.0
Build a regression model predicting the chlorine concentration
given percentage of the watershed covered by roadways. Determine
a) The estimated regression coefficients βˆ0 and βˆ1 .
b) The estimated error variance σ
ˆ2.
c) 95% confidence interval for β1 .
d) 99% prediction interval for a new observation of the chloride concentration when x = 0.64.
e) Perform an F -test for significance of the regression.
f) Find the value of the coefficient of determination.
You can use the following quantities computed from the raw data
in the table above:
x
¯ = 0.515, y¯ = 10.375, Sxx = 0.3856, Sxy = 5.247,
SSE = 12.597, SST = 83.995