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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