Research School of Finance, Actuarial Studies and Statistics
PAST FINAL EXAMINATION 1
STAT7055 Introductory Statistics for Business and Finance
Writing Time: 180 minutes
Reading Time: 15 minutes
Exam Conditions:
Central examination.
Students must return the examination paper at the end of the examination.
This examination paper is not available to the ANU Library archives.
Materials Permitted in the Exam Venue:
(No electronic aids are permitted, e.g., laptops, phones).
Calculator (non-programmable).
Two A4 pages with notes on both sides.
Unannotated paper-based dictionary (no approval required).
Materials to be Supplied to Students:
Script book.
Scribble paper.
Instructions to Students:
Please write your student number in the space provided on the front of the script book.
Attempt all 5 questions.
Start your solution to each question on a new page and clearly label each solution with the corresponding
question number.
To ensure full marks show all the steps in working out your solutions. Marks may be deducted for failure
to show working or formulae.
Selected statistical tables are attached to the back of the examination paper.
If a required degree of freedom is not listed in a statistical table, please use the closest degree of freedom.
Unless otherwise stated, use a significance level of α = 5%.
Round all numeric answers to 4 decimal places.
Question: 1 2 3 4 5 Total
Marks: 19 17 17 9 41 103
Question 1 [19 marks]
(a) [4 marks] A student answers a multiple-choice examination question that offers
four possible answers. Suppose that the probability that he knows the answer to
the question is 0.8 and the probability that he guesses is 0.2. Assume that if the
student guesses, the probability of selecting the correct answer is 0.25. If the student
correctly answers a question, find the probability that he really knew the correct
answer.
(b) [5 marks] Suppose the student takes an examination with 3 multiple choice ques-
tions. Let X denote the number of questions he gets correct. Assuming that the
student answers each question independently, determine the probability distribution
of X and calculate the expected value of X.
(c) [10 marks] Suppose the student takes an examination with 2 multiple choice ques-
tions. But this time, he gains confidence every time he knows the answer to a ques-
tion. Specifically, if he knows the answer to the current question, then for the next
question the probability that he knows the answer becomes 0.9 and the probability
that he guesses becomes 0.1. If he does not know the answer to the current ques-
tion, then for the next question the probability that he knows the answer and the
probability that he guesses remain 0.8 and 0.2, respectively. Again letting X denote
the number of questions he gets correct, determine the probability distribution of X
and calculate the expected value of X. We can assume the probabilities of knowing
the answer and guessing for the first question are 0.8 and 0.2, respectively.
Past Final Examination 1 Page 2 of 7 STAT7055
Question 2 [17 marks]
Suppose that X is a continuous random variable with the following probability density
function:
f(x) =
1
θ
, 0 ≤ x ≤ θ
Jimmy considers himself a budding statistician and he wants to investigate the true value
of θ.
(a) [2 marks] Jimmy thinks that the true value is actually θ = 2. Assuming Jimmy
is right, find each of the following probabilities:
P
(
0 < X <
1
3
)
, P
(
1
3
< X < 1
)
, P
(
1 < X <
7
4
)
, P
(
7
4
< X < 2
)
(b) [4 marks] In order to test whether θ = 2, Jimmy collects a sample of 500 of these
X variables and records their values. He then tabulates how many variables fell
into each range of part (a), and his results are summarised below. Based on this
data, test whether θ = 2. Clearly state your hypotheses and use a significance level
of α = 5%.
Range of value
(
0 < X < 1
3
) (
1
3
< X < 1
) (
1 < X < 7
4
) (
7
4
< X < 2
)
Number of variables 101 165 191 43
(c) [4 marks] Jimmy now wants to actually estimate θ. From the sample he collected
in part (b), he can calculate the sample mean, X¯ =
∑500
i=1Xi
500
. Is X¯ an unbiased
estimator of θ? Why or why not? If not, derive an unbiased estimator of θ.
(d) [3 marks] Jimmy decides it might be a better idea to use an interval estimator
rather than a point estimator. Based on the sample mean of X¯ = 0.9418 and
the population variance of σ2 = 0.3008, calculate a 95% confidence interval for
µ = E(X). Interpret this confidence interval.
(e) [2 marks] Without actually performing the test, if you were to test H0 : µ = 1
against the two-tailed alternative at a significance level of α = 5%, would you reject
H0? Why or why not?
(f) [2 marks] Jimmy actually wanted an interval estimator for θ, not µ. Suggest a way
to convert the confidence interval for µ you constructed in part (d) to a confidence
interval for θ.
Past Final Examination 1 Page 3 of 7 STAT7055
Question 3 [17 marks]
A survey was conducted to determine student, faculty, and administration attitudes on
a new university parking policy. The distribution of those favouring or opposing the
policy was as shown in the table below:
Student Faculty Administration
Favour 252 107 43
Oppose 139 81 40
(a) [4 marks] The parking office claims that the population proportion of people at
the university who are both students and in favour of the new parking policy is
greater than 34%. Test this claim at a significance level of α = 5%, clearly stating
your hypotheses.
For parts (b) and (c), assume that the students, faculty and administration were sampled
independently.
(b) [4 marks] The parking office also claims that the population proportion who favour
the new parking policy is that same for both students and faculty. Test this claim
at a significance level of α = 5%, clearly stating your hypotheses.
(c) [4 marks] The parking office also claims that the population proportion of students
who favour the new policy minus the population proportion of administration who
favour the new policy is equal to 0.1. Test this claim at a significance level of
α = 5%, clearly stating your hypotheses.
(d) [5 marks] The parking office now decides that it is too risky to make claims about
population proportions. Instead, they now only claim that the attitudes regarding
the new parking policy are independent of student, faculty or administration status.
Test this claim at a significance level of α = 5%, clearly stating your hypotheses.
Past Final Examination 1 Page 4 of 7 STAT7055
Question 4 [9 marks]
Suppose the manager of a manufacturing plant suspects that the output of a production
line (in number of units produced per shift) depends on two factors:
• Which of two supervisors is in charge of the line.
• Which of three shifts (day, swing, or night) is being measured.
Suppose that the two supervisors were each observed on three randomly selected days for
each of the three different shifts. The production outputs were recorded and summarised
below:
Shift
Day Swing Night
Supervisor
1 571 480 470
610 474 430
625 540 450
2 480 625 630
516 600 680
465 581 661
(a) [5 marks] Suppose the sample variances of production output for the two super-
visors are s21 = 5155.25 and s
2
2 = 6096.5. Test whether the mean production output
is the same for the two supervisors. Clearly state your hypotheses and use a signifi-
cance level of α = 5%. Assume that the population variances of production output
for the two supervisors are equal.
A two-way ANOVA has been conducted on this data and the results are summarised in
the table below:
Source Sum of squares Degrees of freedom Mean squares F
Supervisor 19208 ? ? ?
Shift 247 ? ? ?
Interaction ? ? ? ?
Error 8640 ? ?
Total 109222 ?
(b) [4 marks] Test whether there is an interaction between supervisor and shift. Clearly
state your hypotheses and use a significance level of α = 5%.
Past Final Examination 1 Page 5 of 7 STAT7055
Question 5 [41 marks]
It is generally thought that the weight of a turkey depends on its age. The following
data was collected on 13 turkeys that were raised on Australian farms:
Age (weeks) Weight (pounds) Age (weeks) Weight (pounds)
28 13.3 26 11.8
20 8.9 21 11.5
32 15.1 27 14.2
22 10.4 29 15.4
29 13.1 23 13.1
27 12.4 25 13.8
28 13.2
Let Y denote Weight and X denote Age. The following summary statistics are provided:
sXY = 5.2404, s
2
X = 12.5769 and s
2
Y = 3.2847.
(a) [5 marks] Fit the regression model Y = β0 + β1X + . That is, calculate the
estimates βˆ0 and βˆ1.
(b) [3 marks] We have been told that the sum of squares for regression is SSR =
26.202. Calculate the standard error of estimate, s.
(c) [4 marks] Test whether there is a linear relationship between Weight and Age.
Clearly state your hypotheses and use a significance level of α = 5%.
(d) [5 marks] Calculate a 95% confidence interval for the expected weight of a turkey
that is 22 weeks old. Would this be wider or narrower than a prediction interval
for the weight of a turkey that is 22 weeks old? Why or why not?
It turns out that the 13 turkeys were not all raised on the same farm. A regression
model with Weight as the dependent variable and Age, NSW and VIC as independent
variables was fitted to the data, where NSW is an indicator variable that equals 1 if the
turkey was raised in New South Wales and VIC is an indicator variable that equals 1 if
the turkey was raised in Victoria. Some regression output is displayed below:
Predictor Coef SE Coef T p-value
Intercept 1.4309 0.6574 2.18 0.0575
Age 0.4868 0.0257 18.91 0.0000
NSW −1.9184 0.2018 −9.51 0.0000
VIC −2.1919 0.2114 −10.37 0.0000
(e) [1 mark] Write down the general form of the model.
(f) [3 marks] Test the overall significance of the model. Clearly state your hypotheses
and use a significance level of α = 5%.
Past Final Examination 1 Page 6 of 7 STAT7055
Analysis of Variance
Source DF SS MS F p-value
Regression 3 38.60575 12.86858 ? ?
Residual Error 9 0.8112 0.0901
Total 12 ?
(g) [3 marks] Calculate the coefficient of determination, R2, and interpret the value.
(h) [4 marks] What is one of the drawbacks of using R2? Calculate the adjusted R2.
(i) [3 marks] What do you conclude about the relationship between Weight and Age?
Clearly state your hypotheses and use a significance level of α = 5%. Interpret the
coefficient parameter for Age.
(j) [2 marks] Test whether a different intercept is needed for turkeys from NSW.
Clearly state your hypotheses and use a significance level of α = 5%.
(k) [2 marks] Test whether a different intercept is needed for turkeys from VIC.
Clearly state your hypotheses and use a significance level of α = 5%.
(l) [5 marks] Suppose we want to test whether a different coefficient parameter for
Age is needed for turkeys from NSW and whether a different coefficient parameter
for Age is needed for turkeys from VIC. What additional variables do you need in
order to test these claims? State clearly the general form of the model that you
would fit and the hypothesis tests you would conduct.
(m) [1 mark] Based on your model from part (l), write down the form of the model
for turkeys from NSW.
END OF EXAMINATION
Past Final Examination 1 Page 7 of 7 STAT7055