Download [24 Marks] Problem 1: [2] (a) Consider tossing a coin 3 times, and

[24 Marks] Problem 1:
[2] (a) Consider tossing a coin 3 times, and define the following events: A = {Toss 3 heads in a row} and
B = {Toss a head, then a tail, then a head}. Choose one of the following answers.
i) P (A) > P (B)
ii) P (A) = P (B)
iii) P (A) < P (B)
iv) Not enough info to tell
[2] (b) Ninety people, 60 males and 30 females, have high blood pressure, and volunteer to be in a study
of a new drug thought to lower blood pressure. The 60 men are randomly divided into three groups of 20,
and the 30 women are randomly divided into three groups of 10. The first of the three groups (20 men, 10
women) is given a placebo, the second group is given a mild dose of the drug and the third group is given a
strong dose of the drug. Neither the researchers, nor the people in the study know which treatment they’ve
been given. Each person has their blood pressure measured before and after being given the treatment.
The researchers record the change in blood pressure for each person, and will compare the changes in blood
pressure for each of the 3 groups.
Fill in each of the blanks below for the experiment just described, using one of the letters from below:
A - Double blind
B - Effectiveness of drug
D - People in the study E - People with high blood pressure
G - Dose of the drug
.
H - Change in blood pressure
——– Experimental Unit
——– Blocking variable
C - Confounding
F - Placebo
I - Gender
——– Factor
——– Response
[2] (c) Suppose that P(A) = 0.47 and P(B) = 0.55, then the events A and B are.
i) Disjoint
ii) Not Disjoint
iii) Not enough info to tell
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[2] (d) Investigators select 257 smokers and 234 non-smokers to be in a study and check in with them 30
years later to see how many from each group have developed lung cancer. It is found that the smoking group
has a much higher proportion of people who have developed lung cancer. Please circle one.
i) The study lacks a control group
ii) We conclude that smoking causes lung cancer
iii) This is a retrospective study
iv) Equal numbers of smokers and non-smokers are required for a valid study
v) None of the above
[2] (e) As part of a quality control program, a company that produces CDs randomly samples one every half
hour and listens to it to determine if the sound quality is good or poor. Let’s define the events:
A = {The first 2 we sample have poor sound quality}
B = {The first 2 we sample have good sound quality}
i) A and B are disjoint
ii) A and B are compliments
iii) A and B are independent
iv) Both i) and ii)
v) None of the above
[2] (f) After a missed season due to a lockout, the NHL has gone through a large set of changes to the rules
in the game of hockey. A newspaper columnist wants to write an article about NHL player’s opinions on the
new rules introduced into the game. He randomly selects 5 NHL teams and then attempts to interview all
of the players on the selected teams. The sampling technique used by the columnist was:
i) Simple Random Sample
ii) Stratified Sample
iii) Cluster Sample
iv) Systematic Sample
v) Convenience Sample
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Use the description of a study below to answer questions (g), (h) and (i) below. In order to assess the opinion
of students at SFU on campus safety, a reporter for the student newspaper interviewed 20 students she met
walking on campus late at night, who were willing to give their opinion. Half of the students said they felt
that campus is safe at night.
[2] (g) The sampling technique used was:
i) Simple Random Sample
ii) Stratified Sample
iii) Cluster Sample
iv) Systematic Sample
v) Convenience Sample
[2] (h) The population of interest was:
i) Students who feel SFU is safe at night
ii) Students who feel SFU is not safe at night
iii) SFU students walking on campus at night
iv) All SFU students
v) None of the above
[2] (i) The true proportion (p) of all SFU students who feel that campus is safe at night is most likely....
i) Close to
1
2
ii) Larger than 21 , since the study is biased
iii) Smaller than 21 , since the study is biased
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[8 Marks] Problem 2:
Roulette is a casino game where a wheel is spun with a ball in it, and the ball lands on one of 38 possible
numbers. 18 of the numbers are black, 18 are red and 2 are green. One bet you can make is on the color
18
of the number that will come up. Suppose you will bet on a black number coming up. Then, there is a 38
chance you will win the bet.
[5] (a) If you decide to make 79 consecutive bets on a black number coming up. What is the probability that
you win more than 50% of the bets you make, and consequently leave the casino with more money than you
came with? Make sure to check the assumptions/conditions of any model you may use.
[3] (b) Suppose you go to the casino three nights in a row, and make 79 bets on a black number coming up
each night. What is the probability that you end up making money on all three nights? That is, what is the
probability that you win more than 50% of the bets made on all three nights?
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[13 Marks] Problem 3:
In the Burnaby area, 42% of adults have a cell phone and 88% have a home phone line. If an adult has a
home phone line, then there is a 36% chance they also have a cell phone.
[3] (a) What is the probability that a randomly selected adult from Burnaby has both a home phone line
and a cell phone?
[3] (b) What is the probability that a randomly selected adult from Burnaby has some sort of phone?
[3] (c) If we know that a randomly selected adult from Burnaby has a cell phone, then what is the probability
that they also have a home phone line?
[2] (d) Are the events ‘owning a cell phone’ and ‘have a home phone line’ independent events? Justify your
answer with some calculations.
[2] (e) Are the events ‘owning a cell phone’ and ‘have a home phone line’ disjoint events? Justify your answer.
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[10 Marks] Problem 4:
SFU is interested in taking a poll of students and asking them the question: “would you recommend coming
to SFU to your friends?”. Students may answer yes or no, and SFU will calculate the proportion of people
who say they would recommend SFU to their friends. The researchers realize that the answer may be
different for people in different areas of study and so they do the following. Suppose that 45% of students at
SFU are in an Arts program, 5% are in a Fine Arts program, 35% are in science and 15% are in commerce.
They will survey 1000 people. They randomly select 450 arts students, 50 fine arts students, 350 science
students and 150 commerce students. Of the 1000 students surveyed, 711 say they would recommend SFU
to a friend.
[2] (a) What type of sampling technique did the researchers use?
[2] (b) What is the population parameter?
[2] (c) What is the sample statistic?
[4] (d) Make and interpret a 95% confidence interval for the true proportion of SFU students who would
recommend coming to SFU to their friends.
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[5 Marks] Problem 5:
It is believed that about 13% of the population is left handed. Researchers are hoping to study a few things
about left-handedness. They will test out all 714 students in a particular school, and they hope to find at
least 82 students who are left-handed to be in their study. What is the probability that they find enough
subjects to be in their study?
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[6 Marks] Problem 6:
A common perception is that some people may be able to delay their death from a chronic illness until
after some special event, like Christmas. Out of 6052 deaths from cancer in either the week before or after
Christmas, 3069 happened in the week after Christmas.
[5] (a) Test the hypothesis that the proportion of deaths coming after Christmas is greater than 50%, using
a significance level of 1%. Make sure to check all assumptions and state a conclusion.
[1] (b) State what the p-value is telling us in the context of this hypothesis test. That is, what is its
interpretation in plain English?
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[8 Marks] Problem 7:
Suppose that you own a shop the sells computers and that 50% of them are manufactured by Hewlett Packard,
30% are manufactured by Dell and 20% are manufactured by Mac. The probability that a computer is found
to be defective is 5% for Hewlett Packard, 8% for Dell and 3% for Mac.
[3] (a) What is the probability that a randomly selected computer from your shop is defective?
[2] (b) If you randomly selected computers in your shop to inspect, how many would you expect to have to
inspect before finding 5 defective ones?
[3] (c) If a randomly selected computer is found to be defective, then what is the probability that it is a
Mac?
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