Research School of Finance, Actuarial Studies and Statistics
PAST FINAL EXAMINATION 4
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: 21 21 18 21 22 103
Question 1 [21 marks]
There are many things which can affect the price of a second hand car. Data was
collected on 105 second hand car sales. A multiple linear regression model was fitted
with sale price as the dependent variable (Y ), and the odometer reading (X1), the
odometer reading squared (X21 ), the age (X2) and an indicator of whether the car has
an automatic transmission (Z = 1 if the car has an automatic transmission and Z = 0
otherwise) as the independent variables. That is, the following model was fitted:
Y = β0 + β1X1 + β2X
2
1 + β3X2 + β4Z +
Note that sale price was measured in thousands of dollars (e.g., Y = 32 corresponds to a
sale price of 32 000 dollars), odometer reading was measured in thousands of kilometres
(e.g., X1 = 19.1 corresponds to 19 100 kilometres) and age was measures in years. The
regression output, which includes some missing entries, is displayed below:
Predictor Coef SE Coef T p-value
Intercept 57.5629 5.7896 9.94 0.0000
Odometer ? ? −2.03 0.0455
Odometer2 ? ? −1.25 0.2136
Age −0.1216 0.1441 −0.84 0.4009
Z −0.1707 0.6040 −0.28 0.7780
Analysis of Variance
Source DF SS MS F p-value
Regression ? ? ? ? ?
Residual Error ? ? ?
Total ? 5783.875
(a) [4 marks] The adjusted R2 for the model is equal to 0.8544375. Test the overall
significance of the model. Clearly state your hypotheses and use a significance level
of α = 5%.
(b) [2 marks] What do you conclude about the relationship between sale price and
odometer reading squared? Clearly state your hypotheses and use a significance
level of α = 5%.
(c) [2 marks] Test whether a different intercept is needed for cars that have an au-
tomatic transmission. Clearly state your hypotheses and use a significance level of
α = 5%.
(d) [3 marks] Test whether the expected change in sale price when the age increases
by 1 year (all other variables held constant) is less than +0.15 (that is, less than
positive 150 dollars). Clearly state your hypotheses and use a significance level of
α = 5%.
Past Final Examination 4 Page 2 of 7 STAT7055
(e) [5 marks] Using the estimated regression model, the predicted sale price for a car
that is 4 years old, travelled 32 000 km (X1 = 32) and has a manual transmission
is yˆ = 22.81569 and the predicted sale price for a car that is 7 years old, travelled
29 000 km (X1 = 29) and has an automatic transmission is yˆ = 26.22919. Based on
this information, calculate the estimates βˆ1 and βˆ2.
When people look to buy second hand cars, the odometer reading is generally the first
thing they check. Given this, a simple linear regression was fitted with sale price (Y )
as the dependent variable and the odometer reading (X1) as the independent variable.
The regression output is given below:
Predictor Coef SE Coef T
Intercept 63.3897 1.5936 39.78
Odometer −1.2906 0.0520 −24.83
(f) [2 marks] Test the overall significance of the model. Clearly state your hypotheses
and use a significance level of α = 5%.
(g) [3 marks] The following sample statistics for the odometer readings are given:
X¯1 = 30.19046 and s
2
X1
= 28.60936. Calculate a 90% prediction interval for the sale
price of a car that has travelled 39 000 km (X1 = 39) given that the standard error
of estimate is s = 2.835572.
Question 2 [21 marks]
The 100 metre sprint is one of the most watched events in any Olympic Games. Sprinters
will often go to great lengths to improve their times, looking into factors such as the
equipment they use and the coaches they hire. Listed in the table below are the personal
best times for the 100 metre sprint for 27 sprinters that are in training. The 27 sprinters
were chosen as follows: For each of three shoe brands (the Fast, the Quick and the
Speedy brand), 9 sprinters who used that particular brand were randomly selected. The
sample variances of the times for each shoe brand are also listed in the table.
Shoe Brand Times s2
Fast 9.82 10.22 10.15 9.77 10.12 9.86 9.87 10.11 10.18 0.0313111
Quick 9.67 9.95 10.10 10.16 9.98 10.09 10.20 10.68 10.29 0.0753278
Speedy 9.69 9.75 9.49 10.03 9.66 10.24 10.04 9.87 10.01 0.0557528
(a) [6 marks] Test whether the mean time for sprinters using the speedy brand is more
than 0.2 seconds faster than for sprinters using the quick brand. Clearly state your
hypotheses and use a significance level of α = 5%. Clearly state any assumptions
you have made (without testing them) when performing this test.
Past Final Examination 4 Page 3 of 7 STAT7055
A one-way ANOVA was performed on this data, using shoe brand as the factor. The
partially filled ANOVA table is provided below:
Source Sum of squares Degrees of freedom Mean squares F
Shoe Brand 0.3059 ? ? ?
Error ? ? ?
Total ? ?
(b) [2 marks] Calculate the sum of squares for error for the one-way ANOVA.
(c) [3 marks] Test whether there is a difference in the mean times between the three
shoe brands. Clearly state your hypotheses and use a significance level of α = 5%.
For each shoe brand, suppose the first 3 sprinters in the table were trained by coach
Andy, the next 3 sprinters were trained by coach Bobby and the last 3 sprinters were
trained by coach Carl. A two-way ANOVA was performed on the same data, using shoe
brand and coach as factors. The partially filled ANOVA table is displayed below:
Source Sum of squares Degrees of freedom Mean squares F
Coach ? ? ? ?
Shoe Brand ? ? ? ?
Interaction 0.2970 ? ? ?
Error 0.6781 ? ?
Total ? ?
(d) [3 marks] Test whether there is an interaction between coach and shoe brand.
Clearly state your hypotheses and use a significance level of α = 5%.
(e) [3 marks] Test whether there is a difference in the mean times between the three
shoe brands. Clearly state your hypotheses and use a significance level of α = 1%.
(f) [4 marks] Test whether there is a difference in the mean times between the three
coaches. Clearly state your hypotheses and use a significance level of α = 1%.
Past Final Examination 4 Page 4 of 7 STAT7055
Question 3 [18 marks]
We have a room of 6 people, all of who are celebrating their birthday today. Each
person’s age (the age that they turned today) is listed below:
31, 18, 21, 21, 18, 29
(a) [3 marks] A study has shown that 70% of people younger than 22 are likely to
hold a party on their birthday, whereas only 22% of people older than 22 are likely
to hold a party on their birthday. For a person randomly selected from the room
of 6 people, find the probability that they will be holding a birthday party today.
Suppose that we select a sample of 2 people from this room (without replacement).
(b) [2 marks] Find the probability that the sample of size 2 contains the oldest person
in the room.
(c) [2 marks] Find the probability that the oldest person in the sample of size 2 is
younger than 24 years old.
(d) [4 marks] Determine the sampling distribution of the age of the oldest person in
the sample of size 2.
(e) [3 marks] Calculate the expected value and variance of the sampling distribution
of the age of the oldest person in the sample of size 2.
(f) [4 marks] Suppose we took 10 samples of size 2, without replacement, from the
room of 6 people (but before each new sample was taken, the previous sample was
returned to the room). Find the probability that in more than 7 samples the oldest
person in the sample was older than 30.
Past Final Examination 4 Page 5 of 7 STAT7055
Question 4 [21 marks]
Let X and Y be independent continuous random variables with the following probability
density functions:
f(x) =

4
9
, 0 ≤ x < 1
−2
9
x+ 6
9
, 1 ≤ x < 2
2
9
, 2 ≤ x ≤ 3
and f(y) =
{
−y + 1, 0 ≤ y < 1
y − 1, 1 ≤ y ≤ 2
(a) [3 marks] Find the probability that X is between 1.2 and 2.2.
(b) [3 marks] Find the probability that X is between 1.2 and 2.2, given that it is
larger than 1.5.
(c) [3 marks] Find the probability that Y is between 0.7 and 1.1 or between 1.2 and
1.9.
(d) [3 marks] Find the probability that X is between 1.2 and 2.2 or Y is between 0.7
and 1.1.
(e) [2 marks] Find the probability that X is between 1 and 1.9, given that Y is
between 0.7 and 1.1.
For parts (f) and (g), we can assume that µX = E(X) =
34
27
and σ2X = V (X) = 0.69204.
(f) [3 marks] Find the probability that, from a sample of size n = 50 taken from the
distribution of X, the sample mean lies between 1.2 and 1.5.
(g) [4 marks] Suppose a sample of size 50 taken from the distribution of Y produced
the following sample statistics:
∑50
i=1 Yi = 44 and
∑50
i=1
(
Yi − Y¯
)2
= 45.08. Based
on this data, test whether the expected value of Y is less than the expected value
of X. Clearly state your hypotheses and use a significance level of α = 5%.
Past Final Examination 4 Page 6 of 7 STAT7055
Question 5 [22 marks]
Suppose X, the amount of time a person stays in bed after their alarm goes off, is
uniformly distributed between 5 and a minutes. Also, suppose Y , the number of minutes
they are late to work is uniformly distributed between 0 and b minutes. Over 100 days,
how long they slept past their alarm (X) and how late they were to work (Y ) were
recorded (in minutes). However, only the number of days for which X and Y fell within
certain ranges was reported in the table below:
5 < X < 7 7 < X < 8 8 < X < 10 Totals
0 < Y < 2 18 9 12
2 < Y < 3 9 4 12
3 < Y < 5 13 9 14
Totals 100
(a) [5 marks] Based on the data given above, test whether a is equal to 10. Clearly
state your hypotheses and use a significance level of α = 5%.
(b) [3 marks] Calculate a 90% confidence interval for the probability (population pro-
portion) that the person sleeps in between 5 and 7 minutes and is between 0 and 2
minutes late for work.
(c) [4 marks] Test whether the probability (population proportion) that the person
is between 2 and 3 minutes late for work is greater than 0.18. Clearly state your
hypotheses and use a significance level of α = 1%.
(d) [5 marks] Based on the data above, test whether X and Y are independent (that
is, whether the rows and columns are independent). Clearly state your hypotheses
and use a significance level of α = 5%.
The following sample statistics were produced from the sample of 100 sleep-in times and
the sample of 100 late-for-work times: X¯ = 7.51092, s2X = 1.894077, Y¯ = 2.488363 and
s2Y = 1.999385. For part (e), assume that the samples of sleep-in times and late-for-work
times are independent, and that σ2X = σ
2
Y .
(e) [5 marks] Test whether a exceeds b by more than 3 minutes. Clearly state your
hypotheses and use a significance level of α = 5%.
END OF EXAMINATION
Past Final Examination 4 Page 7 of 7 STAT7055