Unit8-part2-sample questions
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School
International College of Manitoba *
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Course
1300
Subject
Computer Science
Date
Jun 14, 2024
Type
Pages
57
Uploaded by UltraBadger3374
Sample Final Exam 1
Part A
1. James is taking a computer science course. The mark breakdown is as follows:
•
Midterm Exam 1 – 20%
•
Midterm Exam 2 – 30%
•
Final Exam – 50%
All exams are out of 100 marks. James received scores of 70 on Midterm Exam 1 and
60 on Midterm Exam 2. To receive a final grade of B
+
in the course, a student requires
a final grade of 75% or higher. What is the minimum score James must get on the final
exam in order to receive a final grade of B
+
?
(A) 78
(B) 83
(C) 86
(D) 90
(E) 95
2. A researcher is studying how the height of children during early adolescence is affected
by milk consumption. She plots the heights (in inches) and the milk consumption (in
cups per day) for a sample of children and wants to fit a least squares regression line to
the data. She calculates the following for this sample of children:
•
The correlation between milk consumption and height is 0.4.
•
The mean milk consumption is 4.0 cups per day.
•
The mean height is 60.0 inches.
•
The standard deviation of milk consumption is 1.5 cups per day.
•
The standard deviation of heights is 3.0 inches.
What is the predicted height of a child who consumes 5 cups of milk per day?
(A) 60.4 inches
(B) 60.8 inches
(C) 61.2 inches
(D) 61.6 inches
(E) 62.0 inches
3. A biologist is studying the mercury levels in fish from three different lakes in the area.
She selects random samples of ten fish from each lake. The sample of 30 fish is a:
(A) stratified sample.
(B) simple random sample.
(C) convenience sample.
(D) multistage sample.
(E) systematic sample.
4. The head of the Department of Mathematics would like to determine whether student
performance in an introductory math course differs depending on the professor of the
course and the time of day the course is offered. Two math professors, Dr. Smith and
Dr. Johnson, are each teaching two sections of the same introductory math course this
semester. Dr. Smith teaches sections A01 (at 9:30 a.m.) and A03 (at 1:30 p.m.) and
Dr. Johnson teaches sections A02 (at 9:30 a.m.) and A04 (at 1:30 p.m.). At the end of
the semester, the department head will compare the average grades of the students in
the four sections.
This is an example of:
(A) a completely randomized design with four treatments.
(B) a randomized block design with two blocks and two treatments.
(C) a randomized block design with four blocks and four treatments.
(D) a matched pairs design with two treatments.
(E) an observational study.
2
The next
two
questions (
5
and
6
) refer to the following:
The following five games are scheduled to be played at the World Curling Championships
one morning. The values in parentheses are the probabilities of each team winning their
respective game.
Game 1:
Finland (0.43)
vs.
Germany (0.57)
Game 2:
USA (0.28)
vs.
Switzerland (0.72)
Game 3:
Japan (0.11)
vs.
Canada (0.89)
Game 4:
Denmark (0.33)
vs.
Sweden (0.67)
Game 5:
France (0.18)
vs.
Scotland (0.82)
5. The outcome of interest is the set of winners for each of the five games.
How many
outcomes are contained in the appropriate sample space?
(A) 5
(B) 10
(C) 25
(D) 32
(E) 64
6. In a sports game, the “favourite” is the team with the higher probability of winning and
the “underdog” is the team that is less likely to win. What is the probability that at
least one underdog wins their game?
(A) 0.93
(B) 0.74
(C) 0.80
(D) 0.67
(E) 0.59
The next
five
questions (
7
to
11
) refer to the following:
Suppose we have the following facts about customers buying alcohol at the Liquor Mart:
•
55% buy wine (
W
).
•
40% buy beer (
B
).
•
22% buy wine
and
beer.
•
6% buy wine
and
vodka (
V
).
•
5% buy beer
and
vodka.
•
50% buy beer
or
vodka.
•
2% buy wine and beer and vodka.
7. If we randomly select a customer, what is the probability the customer buys vodka?
(A) 0.10
(B) 0.15
(C) 0.19
(D) 0.25
(E) 0.28
3
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8. Which of the following statements is
true
?
(A) Events
W
and
B
are independent.
(B) Events
W
and
B
are mutually exclusive.
(C) Events
W
and
V
are independent.
(D) Events
W
and
V
are mutually exclusive.
(E) Events
B
and
V
are independent.
9. If we randomly select a customer, what is the probability the customer buys
only
wine?
(A) 0.14
(B) 0.17
(C) 0.25
(D) 0.29
(E) 0.34
10. In a random sample of 17 customers, what is the probability that exactly 10 of them
buy wine?
(A) 0.1153
(B) 0.1367
(C) 0.1495
(D) 0.1629
(E) 0.1841
11. In a random sample of 300 customers, what is the approximate probability that less than
38% of them buy beer?
(A) 0.0808
(B) 0.1492
(C) 0.1736
(D) 0.2389
(E) 0.2929
12. In a particular election, 40% of voters voted for the NDP, 35% voted for the Liberals
and 25% voted for the Conservatives. If we take a random sample of two voters, what
is the probability that they voted for different parties?
(A) 0.765
(B) 0.345
(C) 0.525
(D) 0.655
(E) 0.485
13. There are four patients on the neonatal ward of a local hospital who are monitored by
two nurses. Suppose the probability (at any one time) of a patient requiring attention
by a nurse is 0.3. Assuming the patients behave independently, what is the probability
at any one time that there will not be enough nurses to attend to all patients who need
them? (i.e., what is the probability that at least three patients require attention at the
same time?)
(A) 0.0756
(B) 0.1104
(C) 0.0837
(D) 0.0463
(E) 0.2646
4
14. The yearly rainfall in Vancouver, B.C. follows a normal distribution with standard de-
viation 172 mm. In 20% of years, the city gets over 1200 mm of rain. What is the mean
annual rainfall in Vancouver (in mm)?
(A) 1055.5
(B) 1165.6
(C) 982.6
(D) 1344.5
(E) 1234.4
The next
two
questions (
15
and
16
) refer to the following:
The time
X
taken by a cashier in a grocery store express lane to complete a transaction
follows a normal distribution with mean 90 seconds and standard deviation 20 seconds.
15. What is the first quartile of the distribution of
X
(in seconds)?
(A) 73.8
(B) 85.0
(C) 69.4
(D) 81.2
(E) 76.6
16. What is the probability that the average service time for the next three customers is
between 80 and 100 seconds?
(A) 0.6156
(B) 0.4893
(C) 0.7212
(D) 0.5559
(E) impossible to calculate with the information given
5
The next
two
questions (
17
and
18
) refer to the following:
The amount
X
spent (in $) by customers in the grocery store express lane follows some
right-skewed distribution with mean $24 and standard deviation $15.
17. What is the probability that the average amount spent by the next three customers is
more than $20?
(A) 0.4619
(B) 0.6772
(C) 0.8186
(D) 0.7673
(E) impossible to calculate with the information given
18. What is the approximate probability that the next 40 customers spend less than $1,000
in total?
(A) 0.5199
(B) 0.6064
(C) 0.6628
(D) 0.5784
(E) 0.6331
19. An economist calculates that, in order to estimate the true mean amount spent per year
by Canadians on Christmas presents to within $60 with 95% confidence, she requires
a sample of 90 Canadians.
What sample size would be required to estimate the true
mean amount spent per year by Canadians on Christmas presents to within $20 with
95% confidence?
(A) 10
(B) 30
(C) 156
(D) 270
(E) 810
20. The breaking strength of yarn used in the production of woven carpet material is nor-
mally distributed with standard deviation 2.4 psi. A random sample of 4 specimens of
yarn from a production run is measured for breaking strength, and a confidence interval
for
μ
is calculated to be (128.609, 134.191). What is the confidence level for this interval?
(A) 0.90
(B) 0.95
(C) 0.96
(D) 0.98
(E) 0.99
6
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21. A variable
X
is described by a semi-circular density curve with mean 0 and standard
deviation 0.4, as shown below:
We take a simple random sample of 100 individuals from the semi-circular distribution
and calculate the sample mean ¯
x
. The sampling distribution of
¯
X
is:
(A) approximately normal with mean 0 and standard deviation 0.004.
(B) semi-circular with mean 0 and standard deviation 0.04.
(C) approximately normal with mean 0 and standard deviation 0.4.
(D) semi-circular with mean 0 and standard deviation 0.004.
(E) approximately normal with mean 0 and standard deviation 0.04.
22. We would like to conduct a hypothesis test to examine whether there is evidence that
the true mean amount spent on textbooks by U of M students in one semester is greater
than $400. A random sample of 50 students is selected and the mean amount spent on
textbooks for one semester is calculated to be $430. Assume the population standard
deviation is known to be $165. What is the P-value for the appropriate hypothesis test?
(A) 0.0985
(B) 0.0409
(C) 0.0764
(D) 0.1112
(E) 0.0630
7
23. A man accused of committing a crime is taking a polygraph (lie detector) test.
The
polygraph is essentially testing the hypotheses
H
0
: The man is telling the truth. vs. H
a
: The man is lying.
Suppose we use a 5% level of significance. Based on the man’s responses to the questions
asked, the polygraph determines a P-value of 0.08. We conclude that:
(A) there is insufficient evidence that the man is telling the truth.
(B) there is sufficient evidence that the man is telling the truth.
(C) there is insufficient evidence that the man is lying.
(D) the probability that the man is lying is 0.08.
(E) the probability that the man is telling the truth is 0.08.
24. A statistician conducted a test of H
0
:
μ
= 10 vs. H
a
:
μ >
10 for the mean
μ
of some
normally distributed variable
X
. Based on the gathered data, the statistician calculated
a sample mean of ¯
x
= 12 and concluded that H
0
could be rejected at the 5% level of
significance. Using the same data, which of the following statements
must be true
?
I.
A test of H
0
:
μ
= 10 vs. H
a
:
μ >
10 at the 1% level of significance would also
lead to rejecting H
0
.
II.
A test of H
0
:
μ
= 9 vs. H
a
:
μ >
9 at the 5% level of significance would also
lead to rejecting H
0
.
III.
A test of H
0
:
μ
= 10 vs. H
a
:
μ
6
= 10 at the 5% level of significance would also
lead to rejecting H
0
.
(A) I only
(B) II only
(C) III only
(D) I and II
(E) II and III
25. A sample of 30 vehicles is outfitted with snow tires. The vehicles travel 80 km/h in winter
driving conditions and apply the brakes. The sample mean and standard deviation of
stopping distances for these 30 vehicles are calculated to be 162 metres and 35 metres,
respectively. We would like to test whether the true mean stopping distance differs from
150 metres. At the 1% level of significance, we should:
(A) reject H
0
, since the P-value is between 0.02 and 0.025.
(B) fail to reject H
0
, since the P-value is between 0.025 and 0.05.
(C) reject H
0
, since the P-value is between 0.025 and 0.05.
(D) fail to reject H
0
, since the P-value is between 0.05 and 0.10.
(E) reject H
0
, since the P-value is between 0.05 and 0.10.
8
The next
two
questions (
26
and
27
) refer to the following:
Researchers are studying the impact of alcohol consumption on a person’s ability to
perform a simple task.
Five subjects complete a small puzzle while sober, and then
again after drinking three glasses of wine. The times (in seconds) are shown below, as
well as some summary statistics:
Subject
1
2
3
4
5
mean
std. dev.
Sober (S)
109
132
248
155
202
169.2
55.9
After Wine (AW)
141
128
290
174
260
198.6
72.5
Diff. (
d
= S
-
AW)
-
32
4
-
42
-
19
-
58
-
29
.
4
23.5
Assume that all necessary normality conditions are satisfied. We would like to conduct
a hypothesis test to determine whether there is evidence that the true mean time to
complete a puzzle is longer after consuming three glasses of wine.
26. What are the hypotheses for the appropriate test of significance?
(A) H
0
:
μ
d
= 0 vs. H
a
:
μ
d
<
0
(B) H
0
:
μ
S
=
μ
AW
vs. H
a
:
μ
S
> μ
AW
(C) H
0
:
¯
X
d
= 0 vs. H
a
:
¯
X
d
<
0
(D) H
0
:
μ
d
= 0 vs. H
a
:
μ
d
>
0
(E) H
0
:
¯
X
S
=
¯
X
AW
vs. H
a
:
¯
X
S
<
¯
X
AW
27. The P-value of the appropriate test of significance is:
(A) between 0.01 and 0.02.
(B) between 0.02 and 0.025.
(C) between 0.025 and 0.05.
(D) between 0.05 and 0.10.
(E) between 0.10 and 0.15.
28. You want to estimate the proportion of Canadians who would support a merger of the
federal NDP and Liberal parties to within 0
.
035 with 92% confidence. What sample size
is required?
(A) 625
(B) 784
(C) 466
(D) 816
(E) 553
9
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29. We would like to estimate the true proportion
p
of Manitobans who are bilingual (fluent
in two languages).
We take a random sample of 200 Manitobans and find that 42 of
them are bilingual. What is the margin of error for a 95% confidence interval for
p
?
(A) 0.0565
(B) 0.0353
(C) 0.0693
(D) 0.0288
(E) 0.0404
30. The Acme Car Company claims that less than 8% of its new cars have a manufacturing
defect. A quality control inspector randomly selects 300 new cars and finds that 15 have
a defect. He conducts a hypothesis test to examine the significance of the car company’s
claim. What is the P-value of the appropriate hypothesis test?
(A) 0.0359
(B) 0.0485
(C) 0.0162
(D) 0.0571
(E) 0.0274
10
Sample Final Exam 1
Part B
1. The shower flow rates (in L/min) for a sample of 30 houses are ordered and shown below:
2.2
2.8
3.8
4.2
4.6
5.0
5.7
6.0
6.2
6.2
6.5
6.5
6.7
6.8
7.0
7.2
7.3
7.4
7.4
7.6
7.7
7.9
8.0
8.1
8.5
8.6
8.7
8.8
8.8
11.4
(a) Find the five-number summary for this data set.
(b) Construct an outlier boxplot for this data set.
Show any necessary calculations.
What is the shape of the distribution, excluding outliers?
2. We would like to examine how the age
X
of a certain model of car affects its selling
price
Y
. The age (in years) and price (in $) for a sample of 10 cars of the same make
and model are recorded from the classified ads in the newspaper one weekend. The least
squares regression line is calculated to be ˆ
y
= 18
,
600
-
1
,
700
x
. It is also determined
that 81% of the variation in a car’s price can be accounted for by its regression on age.
(a) Provide an interpretation of the slope of the least squares regression line in this
example.
(b) What is the value of the correlation between age and price for these cars?
(c) One car in the sample is 5 years old and has a price of $15,500. Calculate the value
of the residual for this car.
3. Three hats each contain ten coins. Hat 1 contains two gold coins, five silver coins and
three copper coins. Hat 2 contains four gold coins and six silver coins. Hat 3 contains
three gold coins and seven copper coins. We randomly select one coin from each hat.
(a) The outcome of interest is the colour of each of the three selected coins. List the
complete sample space of outcomes and calculate the probability of each.
(b) Let
X
be the number of gold coins selected. Find the probability distribution of
X
.
4. State the Central Limit Theorem.
5. We take a random sample of 27 trees in a large forest and measure their heights. The
sample mean is calculated to be 28.4 feet and the sample standard deviation is calculated
to be 5.3 feet. Heights of trees in the forest are known to follow a normal distribution.
(a) Construct a 98% confidence interval for the true mean height of all trees in the
forest.
(b) Provide an interpretation of the confidence interval in (a).
(c) Conduct a hypothesis test at the 2% level of significance to determine if there is
evidence that the true mean height of all trees in the forest is less than 30 feet. Show
all of your steps, including the hypotheses, test statistic, P-value and a properly-
worded conclusion.
(d) Provide an interpretation of the P-value of the test in (c).
(e) Could the confidence interval in (a) have been used to conduct the test in (c)? Why
or why not? If it could have been used, explain what the conclusion would be, and
why.
12
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Sample Final Exam 2
Part A
1. The receipt totals (rounded to the nearest dollar) for a sample of 25 customers in the
express lane at a supermarket are ordered and shown below:
5
8
9
13
16
18
19
19
21
23
24
26
30
34
35
37
42
45
48
50
56
56
62
64
73
What is the interquartile range for this data set?
(A) 29
(B) 29.5
(C) 30.5
(D) 31
(E) 32
2. The table below displays the number of pass completions
X
and the number of passing
yards
Y
for six Winnipeg Blue Bomber quarterbacks during the 2015 football season:
Quarterback
1
2
3
4
5
6
Completions
107
34
16
51
149
1
Passing Yards
1434
346
169
453
1757
6
The least squares regression line is calculated to be ˆ
y
=
-
53
.
51 + 12
.
53
x
.
What is the value of the residual for Quarterback 5 (Matt Nichols)?
(A)
-
109
.
97
(B)
-
56
.
46
(C)
-
14
.
46
(D) 14
.
46
(E) 56
.
46
3. A farmer grows pumpkins, whose weights follow a normal distribution with mean 18
pounds and standard deviation 3 pounds. A supermarket will only buy those pumpkins
that weigh between 15 and 25 pounds. What proportion of the farmer’s pumpkins will
the supermarket buy?
(A) 0.7850
(B) 0.8314
(C) 0.8876
(D) 0.9772
(E) 0.9996
4. The yearly rainfall in Regina, Saskatchewan follows a normal distribution with mean 384
mm and standard deviation
σ
. In 10% of years, the city gets less than 320 mm of rain.
What is the standard deviation of the amount of annual rainfall in Regina?
(A) 40 mm
(B) 50 mm
(C) 60 mm
(D) 70 mm
(E) 80 mm
5. From past records, the professor of a large university course has established the following
distribution of grades received by students in the course (with some values missing):
Grade
A
+
A
B
+
B
C
+
C
D
F
Probability
0.08
0.17
???
0.13
???
0.22
0.09
0.07
A student requires a grade of C or better to pass the course. What is the probability
that a randomly selected student passes the course?
(A) 0.76
(B) 0.62
(C) 0.84
(D) impossible to calculate without at least one of the missing probabilities
(E) impossible to calculate without both of the missing probabilities
6. Over the last year, suppose it is known that 32% of Winnipeggers have been to a Jets
hockey game, 25% of Winnipeggers have been to a Blue Bombers football game, and
63% of Winnipeggers have been to
neither
a Jets game nor a Bombers game. What
is the probability that a randomly selected Winnipegger has been to
both
a Jets game
and a Bombers game over the last year?
(A) 0.08
(B) 0.12
(C) 0.15
(D) 0.18
(E) 0.20
14
7. A recently married couple plans to have two children. The outcome of interest is the
gender of each of the two children. Consider the event that exactly one of the couple’s
children will be a boy. Which of the following is the complement of this event?
(A) two boys
(B) two girls
(C) one girl
(D) at least one girl
(E) zero or two girls
The next
three
questions (
8
to
10
) refer to the following:
We have a small deck of ten cards. Five of the cards are red, three are blue and two are
green. We randomly select four cards from the deck
with replacement
. That is, after
we select a card and record the colour, we replace the card in the deck and thoroughly
shuffle it before we select another card.
8. Let
X
be the number of blue cards that are selected. The distribution of
X
is:
(A) binomial with parameters
n
= 4 and
p
= 0
.
1.
(B) binomial with parameters
n
= 10 and
p
= 0
.
3.
(C) binomial with parameters
n
= 10 and
p
= 0
.
4.
(D) binomial with parameters
n
= 4 and
p
= 0
.
3.
(E) not binomial.
9. Let
A
be the event that the first selected card is the only red card in our four selections.
Which of the following events is mutually exclusive of the event
A
?
(A) second card selected is blue
(B) no green cards are selected
(C) third selected card is the only green card
(D) same number of red and blue cards are selected
(E) same number of blue and green cards are selected
10. Now suppose that, instead of selecting four cards, we repeatedly select cards with re-
placement (replacing the card and shuffling the deck after each draw) until we draw a
green card for the first time. What is the probability that we draw our first green card
on the fifth draw?
(A) 0.0819
(B) 0.4096
(C) 0.1342
(D) 0.0554
(E) 0.1746
15
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11. A backpacking party carries five emergency flares, each of which will light with a prob-
ability of 0.93. What is the probability that exactly four of the flares will light?
(A) 0.1271
(B) 0.0524
(C) 0.2947
(D) 0.2618
(E) 0.1835
12. Suppose it is known that 11% of students at a large university live in dorms. In a random
sample of 500 students at the university, what is the approximate probability that at
least 14% of them live in dorms?
(A) 0.0735
(B) 0.0418
(C) 0.0162
(D) 0.0028
(E) 0.0968
13. A variable
X
follows some left-skewed distribution with mean 100 and standard deviation
50. We take a random sample of 2,500 observations from this distribution and create a
histogram of the data values. We expect the histogram to be:
(A) approximately normal with mean close to 100 and standard deviation close to 1.
(B) skewed to the left with mean close to 100 and standard deviation close to 1.
(C) approximately normal with mean close to 100 and standard deviation close to 50.
(D) skewed to the left with mean close to 100 and standard deviation close to 50.
(E) approximately normal with mean close to 100 and standard deviation close to 0.02.
14. A manufacturer of automobile batteries claims that the distribution of battery lifetimes
has a mean of 54 months and a variance of 36 months squared. A consumer group decides
to check the claim by purchasing a sample of 50 of these batteries and subjecting them
to tests to determine their lifetime. Assuming the manufacturer’s claim is true, what is
the probability that the sample has a mean lifetime less than 52 months?
(A) 0.1292
(B) 0.3707
(C) 0.0091
(D) 0.4909
(E) 0.3483
15. A recycling plant compresses aluminum cans into bales. The weights of the bales are
known to follow a normal distribution with mean 100 pounds standard deviation 8
pounds. What is the probability that a random sample of 64 bales has a mean weight
between 99 and 101 pounds?
(A) 0.3413
(B) 0.4772
(C) 0.5561
(D) 0.6826
(E) 0.7485
16
16. We would like to estimate the value of the mean
μ
of some population. Ten statisticians
each take a separate random sample of 100 individuals from the population, and each
of them calculates a 90% confidence interval for
μ
. What is the probability that exactly
eight of their confidence intervals will contain the value of
μ
?
(A) 0.1445
(B) 0.1937
(C) 0.2324
(D) 0.2891
(E) depends on the value of
μ
17. We would like to construct a confidence interval to estimate the true mean systolic blood
pressure of all healthy adults to within 4 mm Hg (millimetres of mercury). We have a
sample of 25 adults available. Systolic blood pressures of healthy adults are known to
follow a normal distribution with standard deviation 8.6 mm Hg. What is the maximum
confidence level that can be attained for our interval?
(A) 80%
(B) 90%
(C) 95%
(D) 96%
(E) 98%
18. Lumber intended for building houses and other structures must be monitored for strength.
A random sample of 25 specimens of Southern Pine is selected, and the mean strength
is calculated to be 3700 pounds per square inch. Strengths are known to follow a normal
distribution with standard deviation 500 pounds per square inch. An 85% confidence
interval for the true mean strength of Southern Pine is:
(A) (3615, 3785)
(B) (3671, 3729)
(C) (3556, 3844)
(D) (3544, 3856)
(E) (3596, 3804)
17
19. The University of Manitoba uses thousands of fluorescent light bulbs each year.
The
brand of bulb it currently uses has a mean lifetime of 900 hours. A manufacturer claims
that its new brand of bulbs, which cost the same as those the university currently uses,
has a mean lifetime of more than 900 hours. It is known that the standard deviation of
lifetimes of the new brand of bulb is 80 hours. The university has decided to purchase
the new brand of bulb if a hypothesis test gives significant evidence supporting the
manufacturer’s claim at the 3% level of significance. A random sample of 64 bulbs were
tested and their mean lifetime was 920 hours. Based on these findings:
(A) the university will not purchase the new brand of bulb, as the P-value is 0.0228.
(B) the university will not purchase the new brand of bulb, as the P-value is 0.0456.
(C) the university will not purchase the new brand of bulb, as the P-value is 0.9772.
(D) the university will purchase the new brand of bulb, as the P-value is 0.0228.
(E) the university will purchase the new brand of bulb, as the P-value is 0.0456.
20. We would like to determine whether the true mean pH level of a lake differs from 7.0.
Lake pH levels are known to follow a normal distribution. We select a sample of ten water
specimens from random locations in the lake. The sample mean pH level is calculated
to be 6.8. A 98% confidence interval for
μ
is calculated to be (6.5, 7.1). Based on this
confidence interval, our conclusion is to:
(A) fail to reject H
0
at the 2% level of significance, since the value 7.0 is contained in
the 98% confidence interval.
(B) fail to reject H
0
at the 1% level of significance, since the value 6.8 is contained in
the 98% confidence interval.
(C) fail to reject H
0
at the 4% level of significance, since the value 7.0 is contained in
the 98% confidence interval.
(D) reject H
0
at the 2% level of significance, since the value 7.0 is contained in the 98%
confidence interval.
(E) reject H
0
at the 4% level of significance, since the value 6.8 is contained in the 98%
confidence interval.
18
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21. We select a random sample of ten oranges from trees grown in a large orchard.
The
standard deviation of the weights of these ten oranges is calculated to be 19 grams.
Based on this sample, a confidence interval for the true mean weight of all oranges
grown in the orchard is calculated to be (126.41, 153.59). What is the confidence level
of this interval?
(A) 90%
(B) 95%
(C) 96%
(D) 98%
(E) 99%
The next
two
questions (
22
and
23
) refer to the following:
Carbon Monoxide (CO) is a colourless and odourless gas. Even at low levels of exposure,
carbon monoxide can cause serious health problems. A home is considered safe if the
mean CO concentration is 4.8 parts per million (ppm) or lower.
We take a random
sample of eight readings in various locations of a home. These readings have a mean of
5.0 ppm and a standard deviation of 0.5 ppm. Assume that CO readings follow a normal
distribution. We would like to conduct a hypothesis test at the 1% level of significance
to determine whether the home is unsafe.
22. The P-value for the appropriate hypothesis test is:
(A) between 0.02 and 0.025.
(B) between 0.025 and 0.05.
(C) between 0.05 and 0.10.
(D) between 0.10 and 0.15.
(E) between 0.15 and 0.20.
23. We conclude that:
(A) there is proof that the home is safe.
(B) there is sufficient evidence that the home is safe.
(C) there is insufficient evidence that the home is unsafe.
(D) there is sufficient evidence that the home is unsafe.
(E) there is insufficient evidence that the home is safe.
19
24. The heights (in inches) of all 24 players on the Winnipeg Jets hockey team are ordered
and shown below:
69
70
70
71
72
72
73
73
73
74
74
75
75
75
75
75
75
75
75
76
77
77
77
80
We would like to determine if the true mean height of Winnipeg Jets hockey players
differs from 75 inches. Which of the following statements is
true
?
(A) We should conduct a
z
test, as it is reasonable to assume heights follow a normal
distribution.
(B) We should conduct a
t
test, as we have no idea of the form of the distribution of
heights.
(C) We should conduct a
z
test, as the sample size is fairly high, so
¯
X
will have an
approximate normal distribution.
(D) We should conduct a
t
test, as the population standard deviation is unknown.
(E) A hypothesis test is unnecessary in this situation.
The next
three
questions (
25
to
27
) refer to the following:
The high school and university GPAs for random samples of students from two univer-
sities are shown below:
University A
Student
1
2
3
4
5
6
7
¯
x
s
High School (H)
3.76
3.64
4.15
4.07
4.20
3.56
3.71
3.87
0.26
University (U)
3.74
4.30
4.09
3.90
3.46
3.37
3.04
3.70
0.44
Diff. (
d
=H
-
U)
0.02
-
0
.
66
0.06
0.17
0.74
0.19
0.67
0.17
0.47
University B
Student
1
2
3
4
5
6
7
8
¯
x
s
High School (H)
3.60
2.79
4.21
3.75
2.58
4.00
4.25
3.14
3.54
0.64
University (U)
3.02
2.42
4.23
3.87
2.52
3.15
4.03
3.00
3.28
0.69
Diff. (
d
= H
-
U)
0.58
0.37
-
0
.
02
-
0
.
12
0.06
0.85
0.22
0.14
0.26
0.33
20
25. We would like to construct a 95% confidence interval for the true mean difference
μ
d
in high school GPA and university GPA for all students at University A. Which of the
following statements are
true
?
(I)
For any University A student, high school GPA and university GPA are
independent.
(II)
For any two University A students, high school GPAs are independent.
(III)
In order to construct the confidence interval, we must assume that differences
in high school GPA and university GPA follow a normal distribution.
(IV)
In order to construct the confidence interval, we must assume that high school
GPAs and university GPAs both follow normal distributions.
(A) I and III only
(B) I and IV only
(C) II and III only
(D) II and IV only
(E) I, II and III only
26. Refer to the previous question. Assuming the necessary normality conditions are satis-
fied, the 95% confidence interval for the true mean difference in high school GPA and
university GPA for all students at University A is:
(A) (
-
0
.
12, 0.46)
(B) (
-
0
.
47, 0.81)
(C) (
-
0
.
03, 0.37)
(D) (
-
0
.
26, 0.60)
(E) (
-
0
.
38, 0.72)
27. We would like to conduct a hypothesis test to determine whether high school GPAs differ
on average from university GPAs for all students at University B. Assuming the proper
normality conditions are satisfied, the P-value for the appropriate test of significance is:
(A) between 0.01 and 0.02.
(B) between 0.02 and 0.025.
(C) between 0.025 and 0.05.
(D) between 0.05 and 0.10.
(E) between 0.10 and 0.15.
21
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28. The North Dakota Department of Tourism is interested in estimating the true proportion
p
of Manitobans visit the state in any given year. What sample size is required in order
to estimate
p
to within 0.06 with 95% confidence?
(A) 234
(B) 245
(C) 256
(D) 267
(E) 278
29. In a random sample of 1,000 university students, 837 of them say they have a Facebook
account.
A 90% confidence interval for the true proportion of all university students
who have a Facebook account is:
(A) (0.818, 0.856)
(B) (0.783, 0.891)
(C) (0.829, 0.845)
(D) (0.794, 0.880)
(E) (0.806, 0.868)
30. In a random sample of 175 university students, 70 have student loans. We would like
to test the claim that more than 30% of all university students have student loans. The
test statistic for the appropriate test of significance is:
(A)
0
.
30
-
0
.
40
r
(0
.
30)(0
.
70)
175
(B)
0
.
40
-
0
.
30
r
(0
.
30)(0
.
70)
175
(C)
0
.
40
-
0
.
30
r
(0
.
35)(0
.
65)
175
(D)
0
.
40
-
0
.
30
r
(0
.
40)(0
.
60)
175
(E)
0
.
30
-
0
.
40
r
(0
.
30)(0
.
40)
175
22
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Sample Final Exam 2
Part B
1. A dentist would like to conduct an experiment to determine the effect of the brand
of toothpaste (Crest or Colgate) and the type of toothbrush (regular or electric) his
patients use on the health of their teeth. The dentist has 240 patients (80 children and
160 adults) who volunteer to participate in the experiment.
The treatments will be
randomly assigned to the subjects. The dentist believes the effect of the treatments will
differ for children and adults, so she conducts a randomized block design.
(a) Identify the following in this experiment:
i. factors
ii. factor levels
iii. treatments
iv. response variable
v. blocking variable
(b) How is control achieved in this experiment?
(c) Suppose at the end of the experiment that it is determined that subjects who used
Crest toothpaste and an electric toothbrush had significantly healthier teeth than
subjects in the other treatment groups. Can we conclude that the treatment was
likely the
cause
?
2. The Winnipeg Jets’ next three scheduled games are shown below, together with the
probabilities that each team will win their respective game, as determined by odds-
makers.
Note that tied games are not possible.
We will assume the outcome of one
game is independent of any other.
Game 1:
Winnipeg Jets (0.6)
vs.
Pittsburgh Penguins (0.4)
Game 2:
Winnipeg Jets (0.7)
vs.
Montreal Canadiens (0.3)
Game 3:
Winnipeg Jets (0.8)
vs.
Buffalo Sabres (0.2)
(a) The outcome of interest is the set of winners of each of the three games. List the
complete sample space of outcomes and calculate the probability of each.
(b) Let
X
be the number of games the Winnipeg Jets win. Find the probability distri-
bution of
X
.
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3. The weights of adobe bricks used for construction follow a normal distribution with mean
5.0 pounds and standard deviation 0.2 pounds.
(a) Only 12% of bricks are heavier than what weight?
(b) We take a random sample of four bricks. Can you calculate the probability that
the average weight of these four bricks is greater than 5.1 pounds? If so, calculate
the probability. If not, explain why not.
(c) What is the probability that the total weight of a random sample of 50 bricks is
less than 248 pounds?
(d) Is the probability you calculated in (c) exact or approximate? Explain.
4. A random sample of 20 students at a large university has a mean GPA of 3.12. GPAs at
the university are known to follow a normal distribution with standard deviation 0.47.
(a) Calculate a 95% confidence interval for the true mean GPA of all students at the
university.
(b) Provide an interpretation of the confidence interval in (a).
(c) Conduct an appropriate hypothesis test at the 5% level of significance to determine
whether there is evidence that the true mean GPA of students at the university
differs from 3.00.
Show all of your steps, including the statement of hypotheses,
the calculation of the appropriate test statistic and P-value, and a carefully-worded
conclusion.
(d) Provide an interpretation of the P-value of the test in (c).
(e) Could the confidence interval in (a) have been used to conduct the test in (c)? Why
or why not? If it could have been used, explain what the conclusion would be, and
why.
24
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Sample Final Exam 3
Part A
1. The five-number summary for the heights (in cm) of the players on a men’s hockey team
is as follows:
162
174
176
181
194
The three goalies on the hockey team are Darren, who is 191 cm tall, Brandon, who
is 165 cm tall, and Kyle, who is 193 cm tall.
If an outlier boxplot for the heights of
the hockey team was constructed, which of these players’ heights would be labelled as
outliers?
(A) only Brandon
(B) only Kyle
(C) Darren and Kyle, but not Brandon
(D) Brandon and Kyle, but not Darren
(E) Darren, Brandon and Kyle
2. Researchers want to determine how the height of a mountain can help explain the tem-
perature at the top of the mountain. The two variables were measured for a sample of
mountains and the least squares regression line was calculated. It was also reported that
58% of the variation in temperature at the top of a mountain can be explained by its
regression on the mountain’s height. What is the value of the correlation between the
two variables?
(A) 0.76
(B) 0.58
(C)
-
0
.
34
(D)
-
0
.
76
(E)
-
0
.
58
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3. How fast do icicles grow?
Researchers measured the growth rate of icicles in a cold
chamber subjected to different temperatures (
-
20
◦
C or
-
30
◦
C) and different wind speeds
(30, or 40 km/h). What is/are the factor level(s) in this experiment?
(A) length of the icicles
(B) growth rate of the icicles
(C) temperature and wind speed
(D)
-
20
◦
C,
-
30
◦
C, 30 km/h, 40 km/h
(E)
-
20
◦
C/30 km/h,
-
20
◦
C/40 km/h,
-
30
◦
C/40 km/h,
-
30
◦
C/40 km/h
4. There are three coloured coins in a hat – one gold coin, one silver coin and one copper
coin. You will randomly select coins from the hat, one at a time without replacement,
until the gold coin is selected, and then you will stop. The outcome of interest is the
sequence of colours that are selected during this process.
How many outcomes are
contained in the appropriate sample space?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 9
5. Consider two events
A
and
B
. We know
P
(
A
) = 0
.
37 and
P
(
A
∪
B
) = 0
.
7354. If
A
and
B
are independent, then what is
P
(
B
)?
(A) 0.47
(B) 0.39
(C) 0.62
(D) 0.44
(E) 0.58
6. A six-sided die has faces numbered 1, 2, 3, 4, 5 and 6. Suppose the die is “loaded”, so
that any particular even-numbered face is twice as likely to land face up as any particular
odd-numbered face.
If we roll this die once, what is the probability that the number
showing is greater than 3?
(A)
1
2
(B)
2
3
(C)
5
9
(D)
7
12
(E)
9
16
26
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7. Two friends are playing a game of Rock, Paper, Scissors.
To play this game, each
player simultaneously and independently selects one of the three items.
From prior
experience, it is known that each of the two friends will select the items with the following
probabilities:
Player
Rock
Paper
Scissors
Joe
0.4
0.2
0.4
Tom
0.5
0.2
0.3
In this game, rock beats scissors, scissors beats paper and paper beats rock. The two
friends will play one game. What is the probability Joe wins against Tom?
(A) 0.30
(B) 0.35
(C) 0.40
(D) 0.45
(E) 0.50
8. Which of the following statements is
false
?
(A) If the probability of
A
is 0.6 and the probability of
B
is 0.5, then
A
and
B
cannot
be mutually exclusive.
(B) If the probability of
A
is 0.4 and the probability of
B
is 0.6, and if
A
and
B
are
independent, then
P
(
A
∩
B
) must be equal to 0.24.
(C) If
P
(
A
) = 0
.
3,
P
(
B
) = 0
.
6, and
P
(
A
∪
B
) = 0
.
72, then
A
and
B
must be indepen-
dent.
(D) If
A
and
B
are mutually exclusive, and if
A
and
C
are mutually exclusive, then
B
and
C
must be mutually exclusive.
(E) Two events that are mutually exclusive cannot be independent.
9. Which of the following variables has a binomial distribution?
(A) You roll five fair dice, each with face values of 1 through 6.
X
= total number of dots facing up on the five dice
(B) You monitor the weather statistics every Saturday in Winnipeg for a year.
X
= number of Saturdays during the year that it snows
(C) A paper boy delivers the newspaper to every house on your block.
X
= number of houses that get their newspaper on time tomorrow morning
(D) You repeatedly flip two quarters simultaneously until both quarters land on Heads.
X
= number of flips required for both quarters to land on Heads
(E) An unprepared student randomly guesses the answer to each of the 30 multiple-
choice questions on the final exam.
X
= number of multiple-choice answers the student gets correct
27
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10. During Tim Hortons’ annual “Roll Up the Rim to Win” promotion, customers who
purchase a cup of coffee check under the rim of the cup to see if they have won a prize.
It is known that 16.7% of all cups are winners. If you buy one cup of coffee every day
for a week (7 days), what is the probability you win at least two times?
(A) 0.1892
(B) 0.2349
(C) 0.2857
(D) 0.3311
(E) 0.3836
The next
three
questions (
11
to
13
) refer to the following:
The amount of soap per bottle for a particular brand of dish soap follows a normal
distribution with mean 828 ml and standard deviation 4 ml.
11. What is the probability that a random sample of 10 bottles of dish soap contain a mean
amount greater than 830 ml?
(A) 0.3085
(B) 0.1539
(C) 0.9429
(D) 0.2296
(E) 0.0571
12. There is an approximate 99.7% chance that a random sample of 16 bottles of dish soap
will contain an average fill volume between:
(A) 825 and 831 ml.
(B) 824 and 832 ml.
(C) 826 and 830 ml.
(D) 816 and 840 ml.
(E) 820 and 836 ml.
13. What amount should be placed on the label of the bottles so that only 4% of bottles
contain less than that amount?
(A) 820 ml
(B) 821 ml
(C) 822 ml
(D) 828 ml
(E) 829 ml
28
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14. When an archer shoots her arrow, she hits the bullseye on the target 78% of the time.
The result of each of her shots is independent of any other. If she shoots 300 arrows,
what is the approximate probability she hits the bullseye at least 240 times?
(A) 0.0162
(B) 0.0436
(C) 0.0719
(D) 0.1401
(E) 0.2005
15. The Central Limit Theorem states that:
(A) when
n
gets large, the sample mean
¯
X
gets closer and closer to the population
mean
μ
.
(B) if a variable
X
has a normal distribution, then for any sample size
n
, the sampling
distribution of
¯
X
is also normal.
(C) if a variable
X
follows a normal distribution, then when
n
gets large, the sampling
distribution of
¯
X
is exactly normal.
(D) when
n
gets large, the standard deviation of the sample mean
¯
X
gets closer and
closer to
σ/
√
n
.
(E) regardless of the population distribution of a variable
X
, when
n
gets large, the
sampling distribution of
¯
X
is approximately normal.
The next
two
questions (
16
and
17
) refer to the following:
The fill volume per bottle for a certain brand of beer follows a normal distribution with
mean 341 ml and standard deviation 3 ml.
16. If you buy a case of 24 bottles of beer, what is the probability that the bottles will
contain an average of exactly 339.5 ml?
(A) 0.0034
(B) 0.0207
(C) 0.0000
(D) 0.0071
(E) 0.0122
17. If you buy a twelve-pack of beer, what is the probability that the total volume of beer
will exceed 4.1 litres?
(A) 0.0838
(B) 0.2206
(C) 0.1397
(D) 0.4443
(E) 0.3632
29
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18. Weights of loaves of bread made at a bakery follow a normal distribution with a mean
of 450 grams and standard deviation 12 grams. A random sample of five loaves of bread
is selected. There is only a 2.5% chance that the average weight of the sample will be
below:
(A) 439.48 grams.
(B) 441.20 grams.
(C) 442.54 grams.
(D) 438.62 grams.
(E) 440.76 grams.
19. Sizes of apartments in a large city follow a normal distribution with mean 846 square
feet and standard deviation 160 square feet. We will take a simple random sample of
four apartments in the city and calculate their average size ¯
x
. The sampling distribution
of
¯
X
is:
(A) approximately normal with mean 846 and standard deviation 40.
(B) exactly normal with mean 846 and standard deviation 40.
(C) approximately normal with mean 846 and standard deviation 80.
(D) exactly normal with mean 846 and standard deviation 80.
(E) impossible to determine with the information given.
20. Consider two Canadian universities.
University A has a student population of 10,000
and University B has a student population of 30,000. A statistician, interested in the
average age of first-year students, collects the general records of samples of 500 first-year
students from University A and 600 first-year students from University B. She assumes
the standard deviations of ages are equal for the two universities. Based on her samples,
she constructs a 95% confidence interval for the true mean age of first-year students at
each university. Which of the following statements is/are
true
?
(I)
The interval for University A will be narrower, as a higher percentage
of students was sampled from University A.
(II)
The interval for University B will be narrower, as a higher number of
students was sampled from University B.
(III)
If the statistician calculated 90% intervals instead, the intervals would be wider.
(IV)
If the statistician doubled her sample sizes, the lengths of her intervals would
be reduced by half.
(A) I only
(B) II only
(C) II and III
(D) I and IV
(E) II and IV
30
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The next
two
questions (
21
and
22
) refer to the following:
City engineers in Hamilton, Ontario would like to estimate the true mean commuting
distance of all workers in the city between home and their principal place of business.
They calculate that, in order to estimate this mean to within
±
1 kilometre with 99%
confidence, they require a sample of 120 workers.
21. What sample size would be required to estimate the true mean commuting distance for
all workers in Hamilton to within
±
2 kilometres with 99% confidence?
(A) 30
(B) 60
(C) 85
(D) 240
(E) 480
22. The city of Montreal has a population five times greater than that of Hamilton (and so it
would be logical to assume that it has five times as many workers). Suppose we wanted
to estimate the true mean commuting distance between home and work for all workers
in Montreal to within
±
1 kilometre with 99% confidence.
Assuming equal standard
deviations for the two cities, we would require a sample of how many Montreal workers?
(A) 24
(B) 120
(C) 269
(D) 600
(E) 3000
23. We take a random sample of
n
individuals and measure the value of some variable
X
. We
conduct a hypothesis test of H
0
:
μ
= 30 vs. H
a
:
μ
6
= 30 at the 5% level of significance.
We calculate a sample mean of ¯
x
= 33 and we reject the null hypothesis. Which of the
following
must
be true?
(I)
A test of H
0
:
μ
= 30 vs. H
a
:
μ
6
= 30 at the 10% level of significance would also
result in the rejection of H
0
.
(II)
A test of H
0
:
μ
= 30 vs. H
a
:
μ >
30 at the 5% level of significance would also
result in the rejection of H
0
.
(III)
A test of H
0
:
μ
= 31 vs. H
a
:
μ
6
= 31 at the 5% level of significance would also
result in the rejection of H
0
.
(A) I only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II and III
31
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24. Prior to distributing a large shipment of bottled water, a beverage company would like
to determine whether there is evidence that the true mean fill volume of all bottles differs
from 600 ml, which is the amount printed on the labels. Fill volumes are known to follow
a normal distribution with standard deviation 2.0 ml. A random sample of 25 bottles is
selected. The sample has a mean fill volume of 598.8 ml and a standard deviation of 3.0
ml. What is the value of the test statistic for the appropriate test of significance?
(A)
t
=
-
0
.
50
(B)
z
=
-
2
.
00
(C)
t
=
-
2
.
00
(D)
z
=
-
3
.
00
(E)
t
=
-
3
.
00
25. We would like to conduct a hypothesis test at the 5% level of significance to determine
whether there is evidence that the true mean amount of jam per jar for a certain brand
of jam differs from 350 ml, the amount stated on the label. We take a simple random
sample of 16 jars and measure the amount of jam in each of them. The sample mean is
calculated to be 348.7 ml. Suppose it is known that the amount of jam per jar follows
a normal distribution with standard deviation 2.8 ml. The correct conclusion is to:
(A) reject H
0
, since the P-value is 0.0157.
(B) reject H
0
, since the P-value is 0.0314.
(C) fail to reject H
0
, since the P-value is 0.0314.
(D) reject H
0
, since the P-value is 0.0628.
(E) fail to reject H
0
, since the P-value is 0.0628.
26. We would like to conduct a hypothesis test of H
0
:
μ
= 50 vs. H
a
:
μ
6
= 50 for the mean
μ
of some normally distributed variable
X
. A random sample of 25 observations is taken
from the population. A 90% confidence interval for
μ
is calculated to be (44
.
46
,
49
.
58).
A 96% confidence interval for
μ
is calculated to be (43
.
82
,
50
.
22). The P-value of the
appropriate test of significance must be:
(A) less than 0.04.
(B) greater than 0.10.
(C) between 0.02 and 0.05.
(D) between 0.04 and 0.10.
(E) equal to 0.06.
32
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The next
three
questions (
27
to
29
) refer to the following:
Coke and Pepsi are the two most popular colas on the market. Do consumers prefer either
one of the two brands of cola over the other? We conduct a matched pairs experiment
as follows: 20 volunteers participate in a blind taste test. Each volunteer tastes both
Coke and Pepsi (in random order) and scores the taste of each cola on a scale from 0 to
100. Some information that may be helpful is shown in the table below:
Scores for Coke
Scores for Pepsi
Difference (
d
= Coke
-
Pepsi)
mean = 78
mean = 83
mean =
-
5
std. dev. = 27
std. dev. = 24
std. dev. = 13
27. Which of the following statements is/are
true
?
(I)
The scores for Coke and Pepsi for each individual are independent.
(II)
The scores for Coke and Pepsi for each individual are dependent.
(III)
In order to conduct the matched pairs
t
test, we must assume that scores for
Coke and scores for Pepsi both follow normal distributions.
(IV)
In order to conduct the matched pairs
t
test, we must assume that differences
in scores (Coke
-
Pepsi) follow a normal distribution.
(A) I only
(B) I and III
(C) I and IV
(D) II and III
(E) II and IV
28. What are the hypotheses for the appropriate test of significance?
(A) H
0
:
μ
d
= 0 vs. H
a
:
μ
d
<
0
(B) H
0
:
¯
X
d
= 0 vs. H
a
:
¯
X
d
6
= 0
(C) H
0
:
μ
C
=
μ
P
vs. H
a
:
μ
C
< μ
P
(D) H
0
:
¯
X
d
= 0 vs. H
a
:
¯
X
d
<
0
(E) H
0
:
μ
d
= 0 vs. H
a
:
μ
d
6
= 0
29. Assuming the appropriate assumptions are satisfied, what is the value of the test statistic
for the appropriate test of significance?
(A)
-
0
.
38
(B)
-
7
.
69
(C)
-
2
.
85
(D)
-
1
.
72
(E)
-
0
.
88
33
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30. Which of the following statements is
false
?
(A) The P-value in a hypothesis test is calculated under the assumption that the null
hypothesis is true.
(B) The lower the P-value of a hypothesis test, the greater the strength of evidence
against the null hypothesis.
(C) The level of significance in a hypothesis test is the highest P-value for which the
null hypothesis will be rejected.
(D) A hypothesis test is designed to assess the evidence in favour of the null hypothesis.
(E) For normally distributed populations, the most important assumption in a hypoth-
esis test for
μ
is that the sample is a simple random sample.
34
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Sample Final Exam 3
Part B
1. Researchers are studying the effect of alcohol consumption on a person’s short-term
memory. Six volunteer subjects are assigned to consume a certain number of drinks
X
.
A short time later, the subjects are given a memory test, and their score
Y
on the test is
recorded. The data are shown below, together with the means and standard deviations:
Subject
1
2
3
4
5
6
mean
std. dev.
No. of Drinks
4
2
8
3
1
6
4
2.61
Test Score
81
90
22
58
80
53
64
25.07
The correlation between number of drinks and test score is calculated to be
r
=
-
0
.
869.
(a) What fraction of the variation in test score can be accounted for by its regression
on number of drinks consumed?
(b) Find the equation of the least squares regression line.
(c) Find the predicted test scores for a subject who consumes 5 drinks and for a subject
who consumes 11 drinks.
(d) Is one of your predictions in (c) more reliable than the other? Explain.
2. Consider the following information about three events
A
,
B
and
C
:
•
P
(
A
) = 0
.
43
•
P
(
C
) = 0
.
28
•
P
(
A
∩
B
) = 0
.
19
•
P
(
B
∩
C
) = 0
.
12
•
P
(
A
∪
B
) = 0
.
61
•
P
(
A
∪
C
) = 0
.
71
(a) What is
P
(
B
)?
(b) Are any two of the three events mutually exclusive?
(c) Are any two of the three events independent?
(d) What is the probability that exactly one of the three events occurs?
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3. We would like to estimate the true mean size
μ
(in square feet) of all two-bedroom
apartments in Winnipeg. A random sample of 30 two-bedroom apartments in the city
is selected, and the mean size of these apartments is calculated to be 1000 square feet.
Suppose it is known that sizes of two-bedroom apartments in the city follow a normal
distribution with standard deviation 200 square feet.
(a) Calculate a 93% confidence interval for the true mean size of all two-bedroom apart-
ments in Winnipeg.
(b) Interpret the meaning of the interval you calculated in (a).
4. An apple grower selects a random sample of 30 apples from this year’s crop. The mean
and standard deviation of weights of these apples are calculated to be 156 grams and 14
grams, respectively. Weights of apples are known to follow a normal distribution.
The apple grower suspects that the true mean weight of apples from this year’s crop
differs from that of last year’s crop, when the true mean weight was 150 grams. Conduct
an appropriate hypothesis test at the 5% level of significance to investigate the grower’s
suspicion. Show all of your steps.
5. We would like to estimate the true proportion
p
of Canadians who support Donald
Trump. In a random sample of 400 Canadians, 80 of them say they support the American
president.
(a) Calculate a 90% confidence interval for the true proportion of Canadians who sup-
port Donald Trump.
(b) Conduct a hypothesis test at the 10% level of significance to determine whether
there is evidence that the true proportion of Canadians who support Donald Trump
is less than 0.25.
(c) Provide an interpretation of the P-value of the test in (b).
36
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Sample Final Exam 4
Part A
1. The average height of all 23 students in a class is 132.5 cm. The average height of the
14 boys in the class is 135.2 cm. What is the average height of the girls in the class?
(A) 129.1 cm
(B) 128.6 cm
(C) 129.8 cm
(D) 127.9 cm
(E) 128.3 cm
2. The batting averages of 34 Major League Baseball players are ordered and shown below:
0.178
0.202
0.210
0.219
0.222
0.237
0.245
0.250
0.256
0.258
0.258
0.260
0.261
0.263
0.267
0.268
0.271
0.275
0.277
0.279
0.279
0.281
0.283
0.284
0.286
0.288
0.294
0.299
0.305
0.311
0.313
0.327
0.338
0.340
The five-number summary is:
0.178
0.256
0.273
0.288
0.340
We construct an outlier boxplot for this data set. To what values do the lines coming
out from the box (i.e., the whiskers) extend?
(A) 0.202 and 0.338
(B) 0.210 and 0.327
(C) 0.208 and 0.336
(D) 0.237 and 0.313
(E) 0.178 and 0.340
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3. A police officer would like to determine how the number of alcoholic beverages consumed
by a person can predict his or her blood alcohol level. The officer measures the values of
both variables on a sample of people leaving a bar one night. The correlation between
the two variables is calculated to be 0.88 and the equation of the least squares regression
line is calculated to be ˆ
y
= 0
.
003 + 0
.
012
x
. Which of the following statements is
true
?
(A) 88% of the variation in blood alcohol level can be accounted for by its regression on
number of alcoholic beverages consumed.
(B) 77% of the variation in number of alcoholic beverages consumed can be accounted
for by its regression on blood alcohol level.
(C) 94% of the variation in blood alcohol level can be accounted for by its regression on
number of alcoholic beverages consumed.
(D) 88% of the variation in number of alcoholic beverages consumed can be accounted
for by its regression on blood alcohol level.
(E) 77% of the variation in blood alcohol level can be accounted for by its regression on
number of alcoholic beverages consumed.
4. An experiment is being conducted to study the effectiveness of different brands of sun-
screen and SPF (sun protection factor) levels. Volunteers will be randomly assigned to
apply either Coppertone or Ombrelle sunscreen, with an SPF level of either 30 or 60.
Subjects will spend eight hours outside in the sun, and the degree of sunburn (if any)
will be compared for all treatments. What is/are the treatment(s) in this experiment?
(A) degree of sunburn
(B) sunscreen brand and SPF level
(C) sunscreen brand, SPF level and degree of sunburn
(D) Coppertone, Ombrelle, SPF 30, SPF 60
(E) Coppertone/SPF 30, Ombrelle/SPF 30, Coppertone/SPF 60, Ombrelle/SPF 60
38
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5. The professor of a large class decides to grade the final exam on a curve. He will use the
following criteria:
•
15% of students will get a grade of A
+
.
•
17% of students will get a grade of A.
•
11% of students will get a grade of B
+
.
•
18% of students will get a grade of B.
•
9% of students will get a grade of C
+
.
•
12% of students will get a grade of C.
•
10% of students will get a grade of D.
•
8% of students will get a grade of F.
Scores on the exam are known to follow a normal distribution with mean 67 and standard
deviation 11. What is the minimum score required to obtain a grade of C?
(A) 56.88
(B) 57.26
(C) 58.90
(D) 59.32
(E) 60.44
6. Labels on the boxes for a certain brand of soap claim the bars of soap weigh 150 grams.
Weights of bars of soap are in fact known to be normally distributed with a standard
deviation of 4 grams. What should be the mean weight of all bars of soap if we want
only 7% of bars to weigh less than the amount stated on the label?
(A) 144.08 grams
(B) 145.83 grams
(C) 153.28 grams
(D) 154.75 grams
(E) 155.92 grams
39
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The next
five
questions (
7
to
11
) refer to the following:
Winnipeg has three professional sports teams – the Winnipeg Jets hockey team, the
Winnipeg Blue Bombers football team and the Winnipeg Goldeyes baseball team.
Suppose we have the following information about Winnipeg residents:
•
55% are Jets fans (J).
•
20% are Goldeyes fans (G).
•
50% are Bombers fans (B) or Goldeyes fans.
•
27% are Jets fans and Bombers fans.
•
11% are Jets fans and Goldeyes fans.
•
10% are Bombers fans and Goldeyes fans.
•
6% are fans of all three teams.
7. What is the probability that a randomly selected Winnipeg resident is a Bombers fan?
(A) 0.30
(B) 0.35
(C) 0.40
(D) 0.45
(E) 0.50
8. Which of the following statements is
true
?
(A) Events J and B are independent.
(B) Events J and G are mutually exclusive.
(C) Events J and G are independent.
(D) Events B and G are mutually exclusive.
(E) Events B and G are independent.
9. What is the probability that a randomly selected Winnipeg resident is a fan of none of
the three teams?
(A) 0.21
(B) 0.23
(C) 0.25
(D) 0.27
(E) 0.29
10. If we take a random sample of 12 Winnipeg residents, what is the probability that exactly
three of them are Goldeyes fans?
(A) 0.2084
(B) 0.2137
(C) 0.2256
(D) 0.2362
(E) 0.2419
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11. If we take a random sample of 500 Winnipeg residents, what is the approximate proba-
bility that less than half of them are Jets fans?
(A) 0.0122
(B) 0.0287
(C) 0.0351
(D) 0.0446
(E) 0.0594
The next
three
questions (
12
to
14
) refer to the following:
We have four boxes, each of which contains 10 coloured balls:
•
Box 1 contains 4 red balls, 5 green balls and 1 yellow ball.
•
Box 2 contains 3 red balls, 5 green balls and 2 yellow balls.
•
Box 3 contains 2 red balls, 5 green balls and 3 yellow balls.
•
Box 4 contains 1 red ball, 5 green balls and 4 yellow balls.
12. If we randomly select one ball from each box, what is the probability of selecting
exactly
one yellow ball?
(A) 0.44
(B) 0.40
(C) 0.36
(D) 0.48
(E) 0.32
13. If we randomly select one ball from each box, what is the probability of selecting
at
least
one yellow ball?
(A) 0.5
(B) 0.6
(C) 0.7
(D) 0.8
(E) 0.9
14. Which of the following variables have a binomial distribution?
(I)
Randomly select three balls from Box 1 with replacement.
X
= number of red balls selected
(II)
Randomly select one ball from each of the four boxes.
X
= number of yellow balls selected
(III)
Randomly select three balls from Box 1 without replacement.
X
= number of red balls selected
(IV)
Randomly select one ball from each of the four boxes.
X
= number of green balls selected
(A) I only
(B) I and III only
(C) I and IV only
(D) II and IV only
(E) I, II and IV only
41
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The next
two
questions (
15
and
16
) refer to the following:
A variable
X
follows a uniform distribution, as shown below:
The mean and standard deviation of this distribution are 3.5 and 2, respectively.
15. We take a random sample of 400 individuals from this distribution and create a histogram
of the data values. We expect this histogram to be:
(A) approximately normal with mean close to 3.5 and standard deviation close to 0.1.
(B) approximately uniform with mean close to 3.5 and standard deviation close to 2.
(C) approximately normal with mean close to 3.5 and standard deviation close to 0.005.
(D) approximately uniform with mean close to 3.5 and standard deviation close to 0.1.
(E) approximately normal with mean close to 3.5 and standard deviation close to 2.
16. We take a random sample of 400 individuals from this distribution and calculate the
value of the sample mean ¯
x
. The sampling distribution of
¯
X
is:
(A) approximately normal with mean 3.5 and standard deviation 0.1.
(B) uniform with mean 3.5 and standard deviation 2.
(C) approximately normal with mean 3.5 and standard deviation 0.005.
(D) uniform with mean 3.5 and standard deviation 0.1.
(E) approximately normal with mean 3.5 and standard deviation 2.
42
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The next
two
questions (
17
and
18
) refer to the following:
Credit card balances for the I. O. U. Credit Corporation follow a normal distribution
with mean $800 and the standard deviation is $200.
17. What is the probability that the mean balance for a random sample of 100 customers is
greater than $780?
(A) 0.7157
(B) 0.5793
(C) 0.9484
(D) 0.8413
(E) 0.6368
18. What is the probability that the total balance for a random sample of 40 customers is
less than $30,000?
(A) 0.0162
(B) 0.0287
(C) 0.0336
(D) 0.0446
(E) 0.0571
19. A variable
X
follows a normal distribution with mean 43 and variance 144. We take a
random sample of
n
individuals from this distribution and calculate the sample mean ¯
x
.
What must the sample size be so that the standard deviation of the sample mean
¯
X
is
equal to 0.8?
(A) 15
(B) 64
(C) 120
(D) 180
(E) 225
20. The number of undergraduate students at the University of Winnipeg is approximately
9,000, while the University of Manitoba has approximately 27,000 undergraduate stu-
dents. Suppose that, at each university, a simple random sample of 3% of the undergrad-
uate students is selected and the following question is asked: “Do you approve of the
provincial government’s decision to lift the tuition freeze?”. Suppose that, within each
university, approximately 20% of undergraduate students favour this decision. What can
be said about the sampling variability associated with the two sample proportions?
(A) The sample proportion for the U of W has less sampling variability than that for
the U of M.
(B) The sample proportion for the U of W has more sampling variability that that for
the U of M.
(C) The sample proportion for the U of W has approximately the same sampling vari-
ability as that for the U of M.
(D) It is impossible to make any statements about the sampling variability of the two
sample proportions without taking many samples.
(E) It is impossible to make any statements about the sampling variability of the two
sample proportions because the population sizes are different.
43
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21. The sizes of farms in a U.S. state follow a normal distribution with standard deviation
30 acres.
We would like to take a sample of farms large enough to estimate the true
mean size of all farms in the state to within 5 acres with 94% confidence. What sample
size is required?
(A) 126
(B) 128
(C) 130
(D) 132
(E) 134
22. A researcher would like to estimate the true mean daily water intake
μ
for all Canadians.
She measures the daily water intake (in ml) for a sample of Canadians and calculates a
95% confidence interval for
μ
to be (1771.6, 1928.4). Daily water intake for Canadians
is known to follow a normal distribution with standard deviation 280 ml. What sample
size did the researcher take?
(A) 16
(B) 25
(C) 36
(D) 49
(E) 64
23. We would like to test whether the true mean IQ of all adult Canadians is less than
110. Suppose that IQs of adult Canadians follow a normal distribution with standard
deviation 17. A random sample of 30 adult Canadians has a mean IQ of 108. What is
the P-value for the appropriate test of H
0
:
μ
= 110 vs. H
a
:
μ <
110?
(A) 0.6444
(B) 0.2090
(C) 0.3556
(D) 0.2611
(E) impossible to determine because the level of significance was not given
24. Which of the following statements comparing the standard normal distribution and the
t
distributions is
false
?
(A) The density curve for
Z
is taller at the centre than the density curve for
T
.
(B) The
t
distributions have more area in the tails than the standard normal distribution.
(C) In tests of significance for
μ
,
Z
should be used as the test statistic when the distri-
bution of
X
is normal, and
T
should be used in other cases.
(D) As the sample size increases, the
t
distributions approach the standard normal
distribution.
(E) In tests of significance for
μ
,
T
should be used as the test statistic when the popu-
lation standard deviation is unknown.
44
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25. A sample of five Major League Baseball games has a mean duration of 168 minutes
and a standard deviation of 17 minutes. Game durations are known to follow a normal
distribution. A 90% confidence interval for the true mean duration of all Major League
baseball games is:
(A) (155.49, 180.51)
(B) (154.18, 181.82)
(C) (153.10, 182.90)
(D) (152.68, 183.32)
(E) (151.79, 184.21)
26. We would like to conduct a hypothesis test at the 10% level of significance to determine
whether the true mean score of all players in a bowling league differs from 150.
The
mean and standard deviation of the scores of 12 randomly selected players are calculated
to be 170 and 16, respectively. Scores of all players in the league are known to follow a
normal distribution. The P-value of the appropriate test of significance is:
(A) between 0.0005 and 0.001.
(B) between 0.001 and 0.002.
(C) between 0.0025 and 0.005.
(D) between 0.005 and 0.01.
(E) between 0.01 and 0.02.
27. For which of the following situations is a matched pairs
t
test appropriate? (Assume all
appropriate normality conditions are satisfied.) A researcher wants to know:
(I)
whether hypnosis reduces pain. 20 volunteers place their hand in ice water and
the time until the pain is unbearable is recorded. They are hypnotized and the
exercise is repeated.
(II)
whether Shell sells more gasoline than Petro Canada. The weekly gas sales are
recorded for samples of 5 Shell gas stations and 5 Petro Canada gas stations.
(III)
whether husbands spend more household money than wives on average. The
amount spent on the husband’s credit card and wife’s credit card are compared
for a sample of 10 married couples.
(A) I only
(B) III only
(C) I and II
(D) I and III
(E) II and III
45
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28. On the evening news, a reporter states, “In a recent poll of likely voters, 50% of re-
spondents indicated that they plan to support the Conservative Party in the upcoming
provincial election. The results are accurate to within 3.1%, 19 times out of 20”. What
sample size was used for the survey?
(A) 600
(B) 800
(C) 1,000
(D) 1,200
(E) 1,400
29. We would like to estimate some population proportion
p
. We take a random sample of
n
individuals and calculate an 80% confidence interval for
p
to be
(0.33, 0.37)
. Using
the same sample, which of the following could be a 99% confidence interval for
p
?
(A) (0.36, 0.38)
(B) (0.32, 0.40)
(C) (0.34, 0.36)
(D) (0.36, 0.44)
(E) (0.31, 0.39)
30. We would like to conduct a hypothesis test at the 5% level of significance to determine
whether the true proportion of Canadians who support Prime Minister Justin Trudeau
differs from 0.50. In a random sample of 300 Canadians, 165 of them say they support
the prime minister. The correct conclusion is to:
(A) reject H
0
, since the P-value is 0.0418.
(B) fail to reject H
0
, since the P-value is 0.0836.
(C) fail to reject H
0
, since the P-value is 0.9582.
(D) reject H
0
, since the P-value is 0.0836.
(E) fail to reject H
0
, since the P-value is 0.0418.
46
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Sample Final Exam 4
Part B
1. A student driving to university must pass through two sets of traffic lights. The first set
of lights is green 50% of the time, yellow 10% of the time, and red 40% of the time. The
second set of lights is green 60% of the time, yellow 10% of the time, and red 30% of the
time. It is also known that the lights function independently.
(a) The outcome of interest is the colour of each of the two lights when the student ar-
rives at the intersections. List the complete sample space of outcomes, and calculate
the probability of each.
(b) What is the probability that both sets of lights are the same colour when the student
arrives at the respective intersections?
(c) What is the probability that the first light is green or the second light is red?
(d) Let
X
be the number of red lights encountered by the student on her way to
university. Find the probability distribution of
X
.
2.
(a) List the four conditions of the binomial setting.
(b) Heights of professional basketball players follow a normal distribution with mean
196 cm and standard deviation 8 cm. What proportion of professional basketball
players are taller than 200 cm?
(c) If we take a random sample of ten professional basketball players, what is the
probability that exactly four of them are taller than 200 cm?
3.
(a) Give the definition of a lurking variable.
(b) Give the definition of a simple random sample.
(c) Explain the difference between a statistic and a parameter. Give three examples of
each.
(d) Explain the difference between two events being mutually exclusive and two events
being independent.
(e) Explain what it means for a result to be statistically significant.
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4. One measure of the quality of education provided by a university is the number of
students per class.
A random sample of 30 third-year classes is selected at a large
university. The number of students in each of the classes are ordered and shown below:
10
11
14
14
15
17
18
20
22
22
22
23
26
26
27
28
31
33
34
34
36
42
44
49
50
55
62
68
77
82
From these data, the sample mean is calculated to be 33.73. The population standard
deviation of class sizes is known to be 18.50.
(a) Construct a histogram for this data set. What is the shape of the distribution of
class sizes for this sample?
(b) We would like to estimate and conduct a hypothesis test for the true mean class size
of all third-year classes at the university. Despite the fact that class size does not
appear to follow a normal distribution, explain why it is nevertheless appropriate
to use inference methods which rely on the assumption of normality.
(c) Construct a 99% confidence interval for the true mean class size of all third-year
classes at this university.
(d) Provide an interpretation of the confidence interval in (c).
(e) Conduct an appropriate hypothesis test at the 1% level of significance to determine
whether there is evidence that the true mean third-year class size at this university
differs from the national average of 40. Show all of your steps.
5. Sixteen people (eight males and eight females) volunteered to be part of an experiment.
All 16 people were between the ages of 25 and 35.
The question of interest in this
experiment was whether females receive faster service at nightclubs than males. Each of
the eight male participants was randomly assigned a bar, and each of the eight females
was randomly assigned to one of these same eight bars. One Friday night, all 16 people
went out to the bar. The male and female assigned to the same bar would arrive within
five minutes of each other. Each person then waited at the bar and ordered a similar
drink. The time (in seconds) until the drink was served was recorded. Some information
that may be helpful is shown below:
Females
Males
Difference (
d
= F
-
M)
mean = 275
mean = 302
mean =
-
27
std. dev. = 37
std. dev. = 48
std. dev. = 29
(a) Conduct an appropriate hypothesis test, at the 1% level of significance. Show all of
your steps, including the statement of hypotheses, the calculation of the appropriate
test statistic and P-value, and a carefully-worded conclusion.
(b) Provide an interpretation of the P-value of the test in (a).
48
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Sample Final Exam Answers
Part A
Sample
Sample
Sample
Sample
Question
Final 1
Final 2
Final 3
Final 4
1
C
C
B
E
2
B
B
D
B
3
A
B
D
E
4
E
B
C
E
5
D
C
E
A
6
C
E
C
E
7
B
E
A
C
8
A
D
D
C
9
D
E
E
D
10
E
A
D
D
11
D
D
E
A
12
D
C
A
A
13
C
D
B
C
14
A
C
E
C
15
E
D
E
B
16
A
B
C
A
17
E
E
B
D
18
C
C
A
E
19
E
D
D
E
20
D
A
B
B
21
E
B
A
B
22
A
D
B
D
23
C
C
B
D
24
B
E
D
C
25
D
C
E
E
26
A
D
D
B
27
B
D
E
D
28
A
D
E
C
29
A
A
D
E
30
E
B
D
B
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Sample Final 1
Part B Solutions
1.
(a) The minimum is 2.2 and the maximum is 11.4.
The median is in position (
n
+ 1)
/
2 = (30 + 1)
/
2 = 15
.
5, so the median is the
average of the 15
th
and 16
th
ordered data values, i.e., (7.0+7.2)/2 = 7.1.
The first quartile is the median of all data values lower in position than the median,
so
Q
1 is in position (15 + 1)
/
2 = 8. Therefore,
Q
1 = 6
.
0.
The third quartile is in position 8 above the median (or equivalently, 8 positions
down from the maximum), and so
Q
3 = 8
.
0.
The five-number summary is:
2.2
6.0
7.1
8.0
11.4
(b) We calculate the fences:
LF
=
Q
1
-
1
.
5
IQR
= 6
.
0
-
1
.
5(8
.
0
-
6
.
0) = 6
.
0
-
3
.
0 = 3
.
0
UF
=
Q
3 + 1
.
5
IQR
= 8
.
0 + 1
.
5(8
.
0
-
6
.
0) = 8
.
0 + 3
.
0 = 11
.
0
There are two values less than the lower fence (2.2 and 2.8) and one value greater
than the upper fence (11.4). These outliers will be plotted as points on the outlier
boxplot.
The whiskers extend to the lowest and highest data values that are not outliers (the
“new minimum” and “new maximum”), i.e., 3.8 and 8.8.
The outlier boxplot is shown below:
Excluding outliers, the distribution is skewed to the left.
50
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2.
(a) When the age of a car increases by one year, we predict its selling price to decrease
by $1,700.
(b) We know that 81% of the variation in a car’s selling price can be accounted for by
its regression on the age of a car. This is the definition of
r
2
, so we know
r
2
= 0
.
81.
The correlation is therefore
r
=
-
√
r
2
=
-
√
0
.
81 =
-
0
.
90
Note that we take the negative square root of
r
2
, since there is a negative association
between the two variables.
(c) The residual for this car is
y
i
-
ˆ
y
i
= 15500
-
(18600
-
1700(5)) = 5400
3.
(a)
S
=
{
GGG, GGC, GSG, GSC, SGG, SGC, SSG, SSC, CGG, CGC, CSG, CSC
}
(b) The probability distribution of
X
is as follows:
x
0
1
2
3
P
(
X
=
x
)
0.336
0.452
0.188
0.024
4. The Central Limit Theorem states that, regardless of the population distribution of a
variable
X
, when the sample size is high (we use
n
≥
30), the sampling distribution of
¯
X
is approximately normal.
5.
(a) (25.87, 30.93)
(b) If we took repeated samples of 27 trees and calculated the interval in a similar
manner, 98% of all such intervals would contain the true mean height of all trees in
the forest.
(c)
t
=
-
1
.
57, 0.05
<
P-value
<
0.10, fail to reject H
0
.
(d) If the true mean height of all trees in the forest was 30 feet, the probability of
observing a sample mean at least as low as 28.4 feet would be between 0.05 and
0.10.
(e) No
51
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Sample Final 2
Part B Solutions
1.
(a)
i. The factors in an experiment are the explanatory variables.
There are two
factors in this experiment – brand of toothpaste and type of toothbrush.
ii. Brand of toothpaste has two factor levels – Crest and Colgate.
Type of toothbrush has two factor levels – regular and electric.
iii. There are 2 x 2 = 4 treatments – Crest/regular, Crest/electric, Colgate/regular
and Colgate/electric.
iv. The response variable is the health of the patients’ teeth.
v. The blocking variable is age (one block consisting of the children and the other
block consisting of the adults).
(b) Control is achieved by the comparison of the four treatment groups in each block.
(Note that blocking is also another form of control. In this case, by using a random-
ized block design, we eliminate the possibility of age becoming a lurking variable.)
(c) Yes. Since this was a properly designed experiment using the principles of random-
ization, replication and control, we can conclude that the treatment was likely the
cause of the observed difference in the response.
2.
(a)
S
=
{
WWW, WWB, WMW, WMB, PWW, PWB, PMW, PMB
}
(b) The probability distribution of
X
is as follows:
x
0
1
2
3
P
(
X
=
x
)
0.024
0.188
0.452
0.336
3.
(a) 5.235
(b) 0.1587
(c) 0.0793
(d) exact
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4.
(a) (2.91, 3.33)
(b) If we took repeated samples of 20 students at the university and calculated the
interval in a similar manner, 95% of such intervals would contain the true mean
GPA of all students at the university.
(c)
z
= 1
.
14, P-value = 0
.
2542, fail to reject H
0
.
(d) If the true mean GPA of all students at the university was 3.00, the probability of
observing a sample mean at least as extreme as 3.12 would be 0.2542.
(e) Yes. Fail to reject H
0
.
53
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Sample Final 3
Part B Solutions
1.
(a) The fraction of the variation in test score that can be accounted for by its regression
on number of drinks consumed is, by definition,
r
2
. Here,
r
2
= (
-
0
.
869)
2
= 0
.
755.
(b) The slope of the least squares regression line is
b
1
=
r
s
y
s
x
=
-
0
.
869
25
.
07
2
.
61
=
-
8
.
347
The intercept of the least squares regression line is
b
0
= ¯
y
-
b
1
¯
x
= 64
-
(
-
8
.
347)(4) = 97
.
388
The equation of the least squares regression line is ˆ
y
= 97
.
388
-
8
.
347
x
.
(c) The predicted test score for a subject who consumes 5 drinks is
ˆ
y
= 97
.
388
-
8
.
347(5) = 55
.
653
(d) The predicted test score for a subject who consumes 11 drinks is
ˆ
y
= 97
.
388
-
8
.
347(11) = 5
.
571
(e) The prediction for the subject who consumed 5 drinks is more reliable, since the
value
x
= 11 falls outside the range of
x
-values in our sample. Predicting the test
score for the subject who consumed 11 drinks is extrapolating.
54
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2.
(a) 0.37
(b) A and B are
not
mutually exclusive.
B and C are
not
mutually exclusive.
A and C
are
mutually exclusive.
(c) A and B are
not
independent.
A and C are
not
independent.
B and C are
not
independent.
(d) 0.46
3.
(a) (933.9, 1066.1)
(b) If we took repeated samples of size 30 and calculated the interval in a similar
manner, 93% of all such intervals would contain the true mean size of all two-
bedroom apartments in Winnipeg.
4.
t
= 2
.
35, 0.02
<
P-value
<
0.04, reject H
0
.
5.
(a) (0.1671, 0.2329)
(b)
z
=
-
2
.
31, P-value = 0
.
0104, reject H
0
.
(c) If the true proportion of Canadians who supported Donald Trump was 0.25, the
probability of observing a sample proportion at least as low as 0.20 would be 0.0104.
55
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Sample Final 4
Part B Solutions
1.
(a)
S
=
{
GG, GY, GR, Y G, Y Y, Y R, RG, RY, RR
}
(b) 0.43
(c) 0.65
(d) The probability distribution of
X
is as follows:
x
0
1
2
P
(
X
=
x
)
0.42
0.46
0.12
2.
(a) A binomial experiment is one for which the following four conditions are satisfied:
•
There is a fixed number of trials
n
.
•
There are only two possible outcomes of interest for each trial – a success or a
failure.
•
The outcome of one trial is independent of any other.
•
The probability of success
p
is constant from trial to trial.
(b) 0.3085
(c) 0.2080
3.
(a) A lurking variable is one that explains the relationship between variables in a study,
but is not included in the study itself.
(b) A simple random sample of size
n
is one that is chosen in such a way that all possible
groups of
n
individuals have the same chance of being chosen.
(c) A statistic is a number that describes a sample.
Examples of statistics include
the sample mean ¯
x
, the sample standard deviation
s
and a sample proportion
ˆ
p
.
A parameter is a number that describes an entire population.
Examples of
parameters include the population mean
μ
, the population standard deviation
σ
and a population proportion
p
.
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(d) Two events that are mutually exclusive have no outcomes in common (i.e., they
cannot both occur at the same time). Two events that are independent can both
occur at the same time, but the outcome of one does not affect the outcome of the
other.
(e) Statistical significance is a difference so large that it would rarely be observed by
chance alone. We say that the results of a hypothesis test are statistically significant
when the P-value
≤
α
and we reject H
0
.
4.
(a) From a histogram of the data, we see the distribution of class sizes for this sample
is skewed to the right.
(b) Although it does not appear that the population distribution of class sizes is nor-
mally distributed, the Central Limit Theorem tells us that since the sample size is
high (we use
n
≥
30), the sampling distribution of the sample mean
¯
X
is approxi-
mately normal.
(c) (25.03, 42.43)
(d) If we took repeated samples of 30 classes and calculated the interval in a similar
manner, 99% of such intervals would contain the true mean class size of all third-year
classes at the university.
(e)
z
=
-
1
.
86, P-value = 0
.
0628, fail to reject H
0
.
5.
(a)
t
=
-
2
.
63, 0.01
<
P-value
<
0.02, fail to reject H
0
.
Note that
n
= 8 (and not 16), as
n
is the number of pairs.
(b) If the service time in nightclubs was the same on average for males and females, the
probability of observing a sample mean difference at least as low as
-
27 would be
between 0.01 and 0.02.
57
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