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Winter 2010
Final Exam 201-301RE
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Marks for each question are included in [ ]on the right, [total marks/80]
Give answers to 2 decimal places unless otherwise noted
1. A political scientist asked a group of people how they felt about two political policy statements. Each
person was to respond A (agree), N (neutral), or D (disagree) to each policy statement.
[4]
a. Describe the sample space; that is, list all possible response combinations to the two statements.
b. Assuming each response combination in the sample space is equally likely, what is the probability
the person being interviewed agrees with at least one of the two policy statements?
c. Assuming each response combination in the sample space is equally likely, what is the probability
the person being interviewed agrees with exactly one of the two political policy statements?
d. Assuming each response combination in the sample space is equally likely, what is the probability
the person being interviewed agrees with the two political policy statements?
2. An interior decorator must furnish two offices. Each office must have a desk, a chair, and a file cabinet . At
a local office furniture store there are 6 models of desks, 8 models of chairs, and 4 models of file cabinet ,
all of which are compatible. (Any desk can be matched with any chair, etc.)
[4]
a. How many choices does the decorator have if he wants to select two desks, two chairs, and two file
cabinets if he doesn’t want to select more than one of any model?
b. How many ways can he hire two technicians with different salaries from a group of 20 all equally
qualified ?
3. Ryan prepares for an exam by studying a list of 20 problems. He can solve 12 of them. For the exam, the
instructor selects ten questions at random from the list of twenty.
[6]
a. In how many ways can 10 problems be chosen from the 12 that Ryan can solve?
b. In how many ways can the 10 problems be chosen from all the 20 problems on the list?
c. What is the probability that Ryan can solve all ten problems on the exam?
4. We must form a four-digit number using the digits 1, 2,3,4,5 without repetitions. How many numbers can
be created?
[2]
5. A group of forty people at a health club were classified according to their gender and smoking habits, as
shown in the table below. One person is selected at random from that group of forty people.
[7]
Smoking Habits
Gender
Smoker (S)
Nonsmoker (N)
Total
Male (M)
2
24
26
Female (F)
6
8
14
Total
8
32
40
a. What is the probability the person does not smoke?
b. What is the probability the person is female?
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Winter 2010
Final Exam 201-301RE
c. What is the probability the person is female and does not smoke?
d. If the person was female, what is the probability she does not smoke.
e. What is the probability the person is female or does not smoke?
6. Lily frequents one of two fast food restaurants, choosing McDonald 25% of the time and Burger King 75%
of the time. If she goes to McDonald, she buys French Fries 10% of the time, and if she goes to Burger
King, she buys French Fries 80% of the time
[4]
a. The next time Lily goes into a fast food restaurant, what is the probability that she goes to
McDonald and orders a French Fries?
b. If Lily goes to a fast food restaurant ,and orders French Fries, what is the probability that she is at
Burger King?
7. A computer repair shop has two work centers. The first center examines the computer to see what is wrong,
and the second center repairs the computer. Let x1 and x 2 be random variables representing the lengths of
time in minutes to examine a computer x1 and to repair a computer x 2 . Assume x1 and x 2 are independent
random variables. Long term history has shown the following times:
Examine computer x1 : 1 =28.1 minutes;  1 =8.2 minutes
; Repair a computer: x 2 :  2 =90.5
minutes;  2 =15.2 minutes
[6]
a. Let W= x1 + x 2 be a random variable representing the total time to examine and repair the computer.
Calculate the mean , variance and standard deviation for the random variable W
b. There is a flat rate of $1.50 per minute to examine the computer, and if no repairs are ordered, there
is also an additional $50 service charge. Let L=1.5 x1 +50. Calculate the mean , variance and
standard deviation for the random variable L
8. Let x denote the weight gain in pounds per month for a calf. The probability distribution of x is shown
below.
[3]
x
0
5
10
15
p(x)
0.1
0.5
0.3
0.1
a. Find the expected average weight gain in pounds per month for a calf?
b. Find the variance of the weight gain.
c. What is P( x  5)?
9. A quiz consists of 5 multiple choice questions. Each question has 5 choices, with exactly one correct
choice. A student, totally unprepared for the quiz, guesses on each of the 5 questions.
[4]
a. How many questions should the student expect to answer correctly?
b. What is the standard deviation of the number of questions answered correctly? Give answer to three
decimal places
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Winter 2010
Final Exam 201-301RE
c. If at least 3 questions must be answered correctly to pass the quiz, what is the chance the student
passes? Give answer to 4 decimal places
10. Roger has read a report that the weights of adult male Siberian tigers have a distribution which is
approximately normal with mean μ = 390 lb. and standard deviation σ = 65 lb.
[3]
a. Find the probability that an individual male Siberian tiger will weigh more than 450 lb.
b. Find the probability that a random sample of 4 male Siberian tigers will have sample mean weight
more than 450 lb.
11. A postal worker has observed that 72% of the customers who buy stamps request particular commemorative
stamps. For a random sample of 80 customers use the Normal distribution with continuity correction to find
the probability that:
[3]
a. 50 or more ask for the special stamps.
b. From 50 to 65 people ask for the commemorative stamps. (Include 50 and 65.)
12. The weights of grapefruit follow a normal distribution. A random sample of 12 new hybrid grapefruit has a
mean weight of 1.7 lb. with sample standard deviation 0.24 lb. Find a 95% confidence interval for the
population mean weight of the hybrid grapefruit.
[2]
13. Air Canada found that 88 out of a random sample of 121 passengers purchased round-trip tickets. Let p be
the proportion of all Air Canada passengers who purchase round-trip tickets. Find a 95% confidence
interval for p.
[2]
14. An overnight package delivery service has a promotional discount rate in effect this week only. For several
years the mean weight of a package delivered by this company has been 10.7 oz. A random sample of 12
packages mailed this week has sample mean weight 11.81 oz with sample standard deviation 2.24 oz. Test
the claim that the mean weight of all packages mailed this week is greater than 10.7 oz. Use a 1%
significance level.
[6]
a.
b.
c.
d.
State the null and the alternate hypotheses.
What is the value of the sample test statistic?
Find (or estimate) the P-value.
State your conclusions in the context of the application.
15. The board of real estate developers claims that 55% of all voters will vote for a bond issue to construct a
massive new water project. A random sample of 215 voters was taken and 96 said that they would vote for
the new water project. Test to see if this data indicates that less than 55% of all voters favor the project. Use
a 1% significance level.
[6]
a.
b.
c.
d.
State the null and the alternate hypotheses.
What is the value of the sample test statistic?
Find (or estimate) the P-value.
State your conclusions in the context of the application.
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Winter 2010
Final Exam 201-301RE
16. A systems specialist has studied the work flow of clerks all doing the same inventory work. Based on this
study, she designed a new work-flow layout for the inventory system. To compare average production for
the old and new methods, a random sample of six clerks was used. The average production rate (number of
inventory items processed per hour) for each clerk was measured both before and after the new system was
introduced. The results are shown in the accompanying table. Assuming that the work rate is normally
distributed, test the claim that the new system speeds up the work rate. Use a 5% significance level. [6]
Clerk
Rate in old system
Rate in new system
a.
b.
c.
d.
1
110
120
2
100
112
3
97
115
4
85
83
5
117
125
6
101
109
State the null and the alternate hypotheses.
What is the value of the sample test statistic?
Find (or estimate) the P-value.
State your conclusions in the context of the application.
17. A study of hypertension involved two groups of men between the ages of 30 and 60. The first group
consisted of a random sample of 42 men who had demanding jobs and control of them, such as executives.
The second group consisted of a random sample of 53 men who also had demanding jobs, but who had little
control over their jobs. In the first group the average systolic blood pressure was 138 with standard
deviation 5. In the second group the average systolic blood pressure was 145 with standard deviation 7. Test
the hypothesis that the mean systolic blood pressure for men in the second group is higher than the mean
systolic blood pressure for men in the first group. Use a 5% level of significance.
[6]
a. State the null and the alternate hypotheses.
b. What is the value of the sample test statistic?
c. Find (or estimate) the P-value.
d. State your conclusions in the context of the application.
18. A lake in northern Quebec was stocked with fish. Seven years later samples were taken to see if the
distribution had changed. Use the following results to test whether the distribution of fish has changed at the
0.01 level of significance.
[6]
a.
b.
c.
d.
Type of fish
Percentage stocked
Bass
Carp
Perch
Trout
30%
25%
5%
40%
State the null and the alternate hypotheses.
What is the value of the sample test statistic?
Find (or estimate) the P-value.
State your conclusions in the context of the application.
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Number of fishes
sampled after seven
years
150
180
30
300
Winter 2010
Final Exam 201-301RE
Answers:
1)
a) The sample space is S = {AA, AN, AD, NA, NN, ND, DA, DN, DD}.
b) 5/9  0.556; c) 4/9  0.444; d) 1/9  0.111
2)
a) Number of choices the decorator has = C26  C28  C24 =(15)(28)(6)=2520; b) P220 =380
12 8
C10
C0
12 8
20
a) C10 C0 ; b) C10 ; c)
20
C10
120
a) P(N) = 32 / 40 = 0.80; b) P(F) = 14 / 40 = 0.35; c)P(F N) = 8 / 40 = 0.20;
d) P(N / F) = P(F N) / P(F) = 0.2 / 0.35 = 0.5714 ; e) P( F  N )  .35  .8  .2  .95
a) Define the following events:
M: Lily chooses McDonald,
B: Lily chooses Burger King, and
F: Lily orders French Fries. Then, P(M) = 0.25, P(B) = 0.75, P(F / M) = 0.10, and P(F / B) =0.8.
Therefore, P(M F) = P(M).P(F / M) = (0.25)(0.1) = 0.025.
b) P(B / F) = P(B F) / P(F) = P(B) . P(F / B) / P(F) = (0.75)(0.8)/0.625=0.96
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
a)  w  28.1  90.5 =118.6minutes ;  w 2  8.2 2  15.2 2 =298.28;  L   w 2 =17.27minutes
b)  L  50  1.5(28.1) =92.15 minutes;  L 2  1.5 28.2 2 =151.29;  L  12.3 minutes
a)    x  p( x) = 7 pounds; b)  2   ( x   )2  p( x) = 16;
c) P( x  5) = P(x = 5) +P(x=10)+P(x=15)= 0.9
a) μ = np = 1; b) σ = npq = 0.894; c) P(x ≥ 3) = 0.0579
a) P(x > 450) = 0.1788; (b) P( x > 450) = 0.0322
a)P(x ≥ 49.5) = P(z ≥ –2.02) = 0.9783;(b) P(49.5 < x < 65.5) = P(–2.02 < z < 1.97) = 0.9539
1.55 lb    1.85 lb
0.65  p  0.81
(a) H0: μ = 10.7 oz; H1: μ > 10.7 oz ; (b) t = 1.7166 ; (c) P value is between 0.05 and 0.075.
(d) Do not reject H0. We cannot conclude that the mean weight of all packages mailed this week is
greater than 10.7 oz.
(a) H0: p = 0.55; H1: p < 0.55; (b) z = –3.05; (c) P value= 0.0011 ;
(d) Reject H0. Fewer than 55% of all voters favor the project.
(a) H0:  d = 0; H1:  d < 0 (  old   new  0) ;(b) t = -3.37; ( d = -9; s = 6.5421)
(c) P value between 0.005 and 0.010; (d) Reject H0. The mean work rate is higher with the new workflow system.
(a) H0: μ 1= μ 2; H1: μ 1< μ 2; (b) t = –5.68 ;(c) P value < 0.0005
(d) Reject H0. The systolic blood pressure for the second group is higher.
(a) H0: The distribution of fish has not changed. H1: The distribution of fish has changed.
(b)  2 = 18.18
(c) p-value < 0.005
(d) Reject H0. The distribution of fish has changed.
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