Download Sample Questions 1. During a space shot, the primary computer

Sample Questions
1. During a space shot, the primary computer system is backed up by two
secondary systems. All the computers act independently and each is
90% reliable. The launch will be successful if at least one of the computers is operating. What is the probability that at least one of the
computers is not functioning and yet the launch is successful?
a) 0
b) 1
c) .667
d) .270
e) none of the preceding
2. The asphalt content of a blend of concrete is normally distributed with
mean µ and variance σ 2. A sample of size 9 is selected and we observe
a sample mean of 5 and a sample variance of 36. Find a 95% confidence
interval for µ.
a) (0.39,9.61)
b) (1.08,8.92)
e) none of the preceding
c) (0.76,9.24)
d) (1.71,8.29)
3. In a sample of 600 families in Ottawa, 360 were found to be renting
their homes, while 240 owned their homes. Find a 95% confidence interval for the proportion of families in Ottawa who own their homes.
a) (.3002, .4008) b) (.6002, .6008) c) (.3671, .4329) d) (.5608, .6392)
e) (.3608, .4392)
4. Air Canada has found that there is a probability of 10% that an individual who has made a reservation will not show up for the flight.
Accordingly, on a small commuter flight with 15 seats, Air Canada
overbooks and accepts 17 reservations What is the probability that all
those who show up for the flight will be able to claim their seats?
a) .8159
b) 1.000
c).5182
d) .4510
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e) none of the preceding
5. My calculator has a function which allows me to generate “random”
numbers between 0 and 1. I generate 50 such numbers and observe that
35 values are greater than .5. Using a level of significance of .01, test
the null hypothesis that the probability of getting a number greater
than .5 less than or equal to 12 , against the alternative than in fact the
probability is larger than 12 . The observed value of the test statistic
and the conclusion are:
a) 2.83, do not reject H0
c) 2.83, reject H0
d) 3.37, reject H0
b) 3.37, do not reject H0
e) none of the preceding
6. An engineer wants to study the relationship between the density x
and hardness y of plywood sheets. Thirty samples are taken, and the
following results are obtained:
X
xi = 7530,
X
x2i = 2155740,
X
yi = 71940,
X
xiyi = 22206047
Estimate the slope in the linear regression model µY |x = α + βx.
a) -67.874
b) 15.615
e) insufficient information given
c) -2.172
d) 5.124
7. Suppose that the length of a plastic strip is normally distributed with
mean 5 cm. and standard deviation .5 cm. What proportion of plastic
strips are not between 4.5 and 5.5 cm. in length?
a) 0.6827
b) 0.5
c) 0.1587
d) 0.8413
e) 0.3173
8. A widget manufacturer has two factories, A and B. 30% of the widgets are made in factory A, and the remainder in B. Suppose that 95%
of the widgets produced by factory A meet specifications while only
85% of the widgets produced by factory B meet specifications. If I buy
a widget made by this manufacturer, what is the probability that it
meets specifications?
a) insufficient information
b) 0.90
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c) 0.95
d) 0.85
e) 0.88
9. Consider the situation described in problem 8. If the widget I buy
meets specifications, what is the probability that it was made in factory A?
a) insufficient information
b) 0.3239
c) 0.5
d) 0.4325
e) 0.7325
10. Suppose that the weight of a female fitness instructor is normally distributed with mean of 55 kg. and standard deviation 5 kg. A group of
36 female fitness instructors is travelling from Sudbury to Ottawa to
attend a seminar. What is the probability that the average weight of
the women in the group is more than 57 kg.?
a) 0.0082 b) 0.9918 c) approximately 0 (< 10−5 ) d) 0.6554 e) 0.3446
11. Let X be a discrete random variable with the following cumulative
distribution function:
x
−1 0
1
2
F (x) 0.2 0.3 0.5 1.0
Find the mean µ = E[X] and the variance σ2 =Var(X) of X.
a) µ = 1.0, σ 2 = 1.4
c) µ = 1.5, σ2 = 0.15
e) µ = 1.5 , σ 2 = 1.0
b) µ = 1.0, σ2 = 2.4
d) µ = 0.5, σ2 = 2.15
12. Let X and Y be independent random variables such that X ∼ Poisson, (2)
and Y ∼ B(5, 0.2). Find
P (X = 0 or Y = 0) = P ({X = 0} ∪ {Y = 0}) .
a) insufficient information b) 0.4630 c) 0.4187 d) 0.4325 e) 0.5621
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13. Suppose that events A1 , A2 , A3 and A4 are mutually exclusive and exhaustive. If P (A1 ) = P (A2) = 0.25 and P (A3 ) = 0.2 , find
P (A2 ∪ A4).
a) 0.65
b) 0.8
c) 0.75
d) 0.55
e) insufficient information
14. Suppose that the requests for service at an information centre follow
a Poisson process with an average of 3 requests per day. What is the
probability that there will be exactly 5 requests in 4 days?
a) 0.0347
b) 0.2573
c) 0.0677
d) 0.1008
e) 0.0127
15. Let X and S 2 be the sample mean and sample variance of a random
sample of size 10 from a N(4, 9) distribution. Find a constant c such
that
Ã
!
X −4
√ ≤ c = .95
P
S/ 10
(a) 1.833
(b) 1.86
(c) 1.645
(d) 2.262
(e) 1.812
16. Suppose that in problem 15, the true mean µ is unknown. If we want
to be at least 95% certain that |X − µ| ≤ 1, what is the smallest sample
size that we should take?
(a) 34
(b) 25
(c) 35
(d) 24
(e) 10
17. The time taken in minutes to complete a particular task in a widget factory follows a normal distribution with mean 30. In an effort
to reduce the mean time required for the task, 16 workers are asked
to carry out the task using a new procedure. The procedure will be
adopted if H0 : µ ≥ 30 is rejected and H1 : µ < 30 is accepted. The
times (in minutes) taken by the 16 workers yield the following data:
x = 29.5, s = 1.2.Find bounds for the p-value of the appropriate test
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statistic, and state your conclusion if α = .05.
(a) .05 < p < .10, do not reject H0
(c) .025 < p < .05, do not reject H0
(e) none of the preceding
(b) .05 < p < .10, reject H0
(d) .025 < p < .05, reject H0
18. An engineer wants to test H0 : µ = 10 versus H1 : µ 6= 10 at level
α = .05, where µ is the mean weight of a steel ingot (in kilograms). If
the weight follows a normal distribution and from experience it is known
that the weight of ingots lies between 6 and 14 kg., what sample size
should be taken in order that β(11) = .1? (β(11) is the probability of
a type II error if µ = 11.)
(a) 36
(b) 42
(c) 44
(d) 54
(e) 53
19. The time taken for a complete recovery was measured for patients who
had undergone one of two different surgical procedures for the same illness. Using the data given below, find a 95% confidence interval for the
difference in the mean recovery time (µ1 − µ2 ) if the two distributions
are independent and normal.
sample size
sample mean
sample variance
Procedure 1 Procedure 2
10
13
7.3 days
8.9 days
1.17 days2
1.43 days2
(a) −1.6 ± 0.31 (b) −1.6 ± 0.99
(e) none of the preceding
(c) −1.6 ± 1.26
(d) −1.6 ± 1.00
20. Suppose that 1% of the radios produced by a certain factory are defective. An engineer tests a random sample of 200 radios. Approximate
the probability that there are at most 3 defective radios in the sample.
a) 0.7996
b) 0.981
c) 0.920
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d) 0.677
e) 0.857
21. Use the information in the following table to test the hypothesis H0
which says that the population means are equal. Find the value of the
F -ratio and state your conclusion at level α = 0.05.
Source
Treatment
Error
Total
Degrees of Freedom
3
Sum of Squares
15953.47
39
16174.50
(a) F-ratio=866.2, reject H0
(b) F-ratio=866.2, do not reject
(c) F-ratio=2.87, reject H0
(d) F-ratio=2.87, do not reject
(e) none of the above
22. Students who are candidates for an honor society are classified according to sex and academic year in the following table:
First year
Second year
Total
Male
16
Female
Total
30
40
80
Test the null hypothesis of “no association” between sex and academic
year. (a) What is the value of the test statistic? (b) What is the
P-value?
My answers are: (a)
(b)
23. In a certain region, one in 1000 women (aged between 50 and 59) will
develop osteoporosis in the interval of one year. What is the probability that at least 4 women in 1000 will develop osteoporosis next year?
(a) 0.019
(b) 0.001
(c) 0.05
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(d) 0.99
(e) none of the above
24. Suppose events A and B are such that P [A] = 0.1 and P [B] = 0.2.
Find P [A or B] in the following cases:
(a) A and B are mutually exclusive;
(b) A and B are independent.
My answers are: (a)
(b)
25. George, John, Paul and Sam are school children that line up daily for
lunch. (a) What is the total number of ways they can arrange themselves in a line? (b) What is the total number of ways they can arrange
themselves in a line such that Paul and George are next to each other?
My answers are: (a)
(b)
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