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FACULTY OF SCIENCE AND AGRICULTURE
AUTUMN SESSION EXAMINATION 2002
QBM 117 BUSINESS STATISTICS
SUBJECT CONVENOR:
Kerrie Cullis (Wagga Wagga)
DAY & DATE:
TIME:
WRITING TIME:
Three (3) hours
MATERIALS SUPPLIED BY UNIVERSITY:
READING TIME:
minutes
Ten (10)
1 x 12 page examination answer
booklet
1 x General Purpose Answer
Sheet
MATERIALS PERMITTED IN EXAMINATION: Battery operated calculator (no
printer)
2B Pencil, eraser, ruler
Text: Australian Business
Statistics by Selvanathan and
Selvanathan, distance education
materials and any other written
materials.
INSTRUCTIONS TO CANDIDATES:
1.
2.
3.
4.
5.
Enter your name and student number and sign in the space provided at the
bottom of this page.
This examination is open book.
This examination consists of two parts.
Part A: 4 Objective Questions
Part B: 20 Multiple Choice Questions
Part A is to be answered in the examination answer booklets provided.
Number each question clearly. Write your name and student number on the
front cover of the answer booklets used.
Part B is to be answered on the General Purpose Answer Sheet, using a 2B
pencil ONLY. Fill in your name and student number. Make sure you fill the
circle completely and make no stray marks on the answer sheet.
This examination is worth 60% of the final assessment.
INSTRUCTIONS TO INVIGILATORS:
1.
The examination paper must not be retained by the candidate.
STUDENT NAME:
STUDENT NO:
STUDENT SIGNATURE:
QBM117 Exam - Autumn 2002
Page1 of 13
PART A
These questions are to be answered in the answer booklet provided.
Question 1
a.
The following frequency table shows the speeds (km/h) of a sample of cars
travelling past a school crossing between 8:30 and 9:30 am on weekdays.
Speeds (km/h)
>20 up to and including 30
>30 up to and including 40
>40 up to and including 50
>50 up to and including 60
>60 up to and including 70
>70 up to and including 80
i.
Frequency
3
24
16
11
5
1
How many cars were sampled?
(1 mark)
ii.
What is the modal class for these data?
(1 mark)
iii.
Sketch a cumulative relative frequency polygon (an Ogive) for these
data. Show all workings and the sketch in your answer booklet. Don't
forget to label all axes and include an informative title.
(7 marks)
iv.
Calculate the mean and standard deviation speed travelled by these
cars.
(3 marks)
v.
Why are the values for part iv. only approximations?
(2 marks)
QBM117 Exam - Autumn 2002
Page2 of 13
b.
A lecturer was interested to compare the grades of students in three different
tutorial classes with three different tutors. The boxplot showing the grade
distribution for each of the three tutorial groups follows.
Grade Distributions for 3 Tutorial classes
in STAT101, 1998
Tutorial
A
B
C
0
20
40
60
80
100
Final Grade (/100)
i.
Which tutorial class had a grade distribution that was the closest to a
normal distribution? How did you determine this?
(3 marks)
ii.
The overall pass mark for the subject was a final grade of 50/100 and
75% of all students enrolled passed. Was the performance of any of
the above tutorial classes inconsistent with the performance of the
group overall? Explain.
(3 marks)
QBM117 Exam - Autumn 2002
Page3 of 13
Question 2
a.
b.
The time taken to install a new aircraft engine is a normally distributed
random variable with a mean of 20 hours and a standard deviation of 1 hour.
i.
What is the probability that the next installation takes between 17 and
18 hours?
(4 marks)
ii.
What is the probability that the next installation takes more than 16.5
hours?
(4 marks)
iii.
A random sample of 10 engines is selected. What is the probability that
the mean time to install the engines is below 19.5 hours?
(5 marks)
The standard medical treatment for a certain disease is successful in 60% of
all cases. The treatment is given to 20 patients.
i.
Let X = the random variable of interest for this problem. Define X.
(1 mark)
ii.
What type of random variable is X?
(1 mark)
iii.
What is the probability that the treatment is successful for less than 10
of the patients?
(2 marks)
iv.
What is the probability that the treatment is successful for 14 or more
of the patients?
(3 marks)
QBM117 Exam - Autumn 2002
Page4 of 13
Question 3
a.
A company is considering installing a fax machine at one of its offices. As part
of the decision process as to whether to install the machine, the company's
manager wants to estimate the average number of documents that would be
transmitted daily if the machine were installed. From experience at other
offices, the company manager believes the standard deviation of the number of
documents sent daily is 32. The manager also believes the number of
documents transmitted daily is a normally distributed random variable. The
machine is tested over a random sample of 15 days, and the resulting sample
mean is 267.
i.
Find a 99% confidence interval estimate for the average number of
documents that would be transmitted daily if the machine were
installed.
(5 marks)
ii.
Suppose the manager decides to install the machine if she could be
fairly confident that the average number of documents transmitted
daily would be above 245. Do the findings in part i. justify installing
the machine? Explain.
(3 marks)
b.
The average total daily sales at a small supermarket are known to be $4528.
The store's management recently implemented some changes in displays of
goods, order within aisles and other changes, and management now wants to
know whether the average sales volume has changed. A random sample of 12
days shows the average sales to be $5019 with a standard deviation of $630. If
sales volume is known to be normally distributed, test whether there has been
a significant change in sales volume. Use  = 0.05.
(7 marks)
c.
Ron Jones, the general manager of the National Paper Company, wants to
determine the mean diameter of pine trees on land that is being considered for
purchase. Past data suggests that the standard deviation of the trees on this
land is 6.35 cm. If Ron wants to estimate the mean diameter to within 1 cm
with 95% confidence, how many trees should be included in the sample?
(5 marks)
QBM117 Exam - Autumn 2002
Page5 of 13
Question 4
a.
The manager of a retail business suspects that people other than customers are
using the store's carpark. She wants to test, at a significance level of   0.05
whether cars in the carpark are typically parked for less than an hour, as those
of customers would be. A random sample of 50 cars shows a mean parking
time of 65 minutes, with a standard deviation of 21 minutes. Make the test.
(6 marks)
b.
A sample of used car salespeople was taken and their Annual gross salary
($000's) and Annual sales turnover ($000's) recorded. A simple linear
regression analysis was performed and the following Excel output generated.
Annual gross salary ($ 000's)
Scatterplot of turnover vs salary
60
55
50
45
40
35
30
250
300
350
400
450
Annual sales turnover ($ 000's)
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.71796444
R Square
0.515472937
Adjusted R Square
0.505378623
Standard Error
2.894990367
Observations
50
ANOVA
df
Regression
Residual
Total
Intercept
Turnover
1
48
49
SS
MS
F
427979841.1 427979841.1 51.06567386
402286522.8 8380969.225
830266363.9
Coefficients
Standard Error
t Stat
P-value
14.43844231
3.812199883 3.787430553 0.000423927
0.078741919
0.011018982 7.146025039
4.393E-09
QBM117 Exam - Autumn 2002
Page6 of 13
i.
What is the equation for the relationship between turnover and salary?
(2 marks)
ii.
What is the increase in gross salary for a one thousand dollar increase
in sales turnover?
(1 mark)
iii.
What is the annual salary of a salesperson who sells no cars?
(1 mark)
iv.
What is the predicted annual salary of a salesperson whose annual sales
turnover is $450 000?
(2 marks)
v.
Find the 95% prediction interval for the annual gross salary of a
salesperson whose turnover is $450 000 in a year? These values may
be of assistance, x  344.966 SS x  69025.780
(4 marks)
vi.
Is the relationship between gross salary and sales turnover significant?
Use a significance level of 0.01.
(4 marks)
QBM117 Exam - Autumn 2002
Page7 of 13
PART B
These questions are to be answered on the General Purpose Answer sheet provided.
Use a 2B pencil only. DO NOT use a blue or black biro.
100 people in a regional town were surveyed. They were asked whether they agreed
with a new development which was being proposed for the main business area. The
results of this survey were presented in graphical form. Use the following graph to
answer questions 1. through 3. inclusive.
Results of a survey re a new development
0.35
0.3
0.25
0.2
0.15
0.1
0.05
1.
is
ag
re
e
ag
re
e
D
y
tr o
ng
l
D
is
to
D
is
S
ag
re
e
gr
ee
to
Te
nd
Midpoints.
Frequency.
Relative frequency.
Number of people.
Height.
The distribution is
A.
B.
C.
D.
E.
3.
Te
nd
A suitable label for the vertical axis would be
A.
B.
C.
D.
E.
2.
A
gr
ee
A
S
tr o
ng
l
y
A
gr
ee
0
multimodal.
bimodal.
peaked.
approximately normal.
unimodal.
The variable displayed is measured at which of the following levels
A.
B.
C.
D.
E.
Nominal.
Ordinal.
Interval.
Ratio.
Continuous.
QBM117 Exam - Autumn 2002
Page8 of 13
4.
A component bar chart should be used if the aim is to
A.
B.
C.
D.
E.
5.
When a distribution is skewed to the right, which of the following statements
is most correct?
A.
B.
C.
D.
E.
6.
median < mean.
mode = median.
The median a better descriptor of central tendency.
All of the above.
A. and C. above.
When quoting average house prices, the median is generally quoted instead of
the mean. This is because
A.
B.
C.
D.
E.
7.
compare relative frequencies.
compare raw frequencies.
compare cumulative frequencies.
compare subjective frequencies.
calculate the mean and standard deviation.
the median is simpler to calculate.
the median is always close to the mean anyway.
house prices are usually from a negatively skewed distribution.
all of the above.
none of the above.
Given that z is the standard normal random variable find P(z > -1.24)
A.
B.
C.
D.
E.
0.3925
0.1075
0.8925
0.6075
0.7850
QBM117 Exam - Autumn 2002
Page9 of 13
Use the following information to answer questions 8. and 9.
An investment analyst collects data on stocks, and notes whether or not dividends
were paid and whether or not the stocks increased in price over a given period. Data
are presented in the following table.
Dividends paid
No dividends paid
Total
8.
0.136
0.286
0.304
0.476
0.924
0.136
0.286
0.304
0.476
0.924
If P(A) = 0.25 and P(B) =0.65, then P(A and B) is
A.
B.
C.
D.
E.
11.
Total
112
138
250
Given that a stock has increased in price, what is the probability that it also
paid dividends?
A.
B.
C.
D.
E.
10.
No price increase
78
53
131
If a stock is selected at random, what is the probability that it both increased in
price and paid dividends?
A.
B.
C.
D.
E.
9.
Price increase
34
85
119
0.1625
0.25
0.40
0.90
unable to be determined due to insufficient information.
The number of accidents that occur weekly on a busy stretch of highway is an
example of
A.
B.
C.
D.
E.
a continuous probability distribution.
a discrete probability distribution.
a Poisson probability distribution.
a continuous random variable.
a discrete random variable.
QBM117 Exam - Autumn 2002
Page10 of 13
12.
Given that z is the standard normal random variable, what is the value of z if
the area to the right of z is 0.1949?
A.
B.
C.
D.
E.
13.
0.51
-0.51
0.86
-0.86
unable to be determined due to insufficient information.
A news television programme asks viewers to phone or fax in their
agreement/disagreement with the expulsion of school pupils for using an
illegal substance at school. Suppose that the television station received 10 000
replies. Which of the following is true?
A.
B.
C.
D.
E.
The 10 000 replies represents a large random sample.
The large sample will provide results that are representative of the
population.
The results are likely to be biased.
All of the above.
None of the above.
Use the following information to answer questions 14. and 15.
The owner of a manufacturing company claims that 30% of its employees earn more
than $40 000 pa. The employees dispute this, arguing that the proportion is in fact
much less than this.
14.
The appropriate hypotheses to test this claim would be
A.
B.
C.
D.
E.
15.
H 0 : p  0.30
H A : p  0.30
H 0 : p  0.30
H A : p  0.30
H 0 : p  0.30
H A : p  0.30
H 0 :   $40 000
H A :   $40 000
H 0 :   $40 000
H A :   $40 000
A random sample of 30 employees were selected, 8 were found to have an
income which exceeded $40 000. The appropriate hypothesis test was
performed using Excel and the following output generated.
Test of Hypothesis About p
Test of p = 0.3 Vs p less than 0.3
Sample Proportion = 0.2667
Test Statistic = -0.3984
P-Value = 0.3452
QBM117 Exam - Autumn 2002
Page11 of 13
Use the output provided to determine which of the following statements is
correct at a 5% level of significance.
16.
A.
B.
Reject H 0 since p-value < 0.05
Reject H 0 since z sample   1.645
C.
Reject H 0 since z sample   1.96
D.
E.
Do not reject H 0 since p-value < 0.5
Do not reject H 0 since z sample   1.645
Which of the following statements about the correlation coefficient between x
and y is true?
A.
B.
C.
D.
E.
17.
The least squares method for fitting a regression line minimises the
A.
B.
C.
D.
E.
18.
It detects whether y is caused by x.
It provides a measure of the linear association between y and x.
It tells us how much y increases for a unit increase in x.
It allows the response variable y to be predicted from the explanatory
variable x.
It tells us how well the regression line fits the data.
standard deviation.
residual sum of squares.
sum of squares of fitted values.
sum of absolute deviations between actual and fitted values.
value of the slope.
In regression, a residual is defined as
A.
B.
C.
D.
E.
the horizontal distance between a point and the regression line.
the distance between consecutive points on the scatterplot.
the variation you would expect if you fitted another variable.
the vertical distance between a point and the regression line.
none of the above.
QBM117 Exam - Autumn 2002
Page12 of 13
19.
From a regression analysis of a response variable on a single explanatory
variable, a plot of residuals against fitted values follows.
Residual plot
Residuals
4
2
0
-2
0
10
20
30
40
50
-4
Fitted values
This plot tells you
A.
B.
C.
D.
E.
20.
there is no relationship between the two variables.
the relationship is not linear.
the wrong explanatory variable has been used.
the constant variation assumption does not hold.
a mistake has been made in the analysis.
We want to predict sales of a product from orders taken. A straight line
regression is fitted, with sales as the response variable and orders taken as the
explanatory variable. Which of the following statements is false?
A.
B.
C.
D.
E.
It is dangerous to use a regression line to predict sales when the
number of orders is outside the range of values used to fit the line.
The intercept in the regression equation is the value of orders when
sales is equal to zero.
A non linear relationship between sales and orders taken can be
detected from a plot of residuals against fitted values.
A. and C. are both false.
B. and C. are both false.
QBM117 Exam - Autumn 2002
Page13 of 13