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FACULTY OF SCIENCE AND AGRICULTURE
SPRING SESSION EXAMINATION 2001
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
May be highlighted and have
notes made in it.
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 to the Selvanathan textbook only.
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 - Spring 2001
Page 1 of 13
PART A
These questions are to be answered in the answer booklet provided.
Question 1
Automobile insurance companies take many factors into consideration when setting
rates including the distance travelled each year. In order to determine the effect of
gender, a sample of 100 male and 100 female drivers were asked how many
kilometres he or she drove in the past year. A frequency distribution and histogram of
the distances (in thousands of kilometres) generated by MS Excel ® for the male
drivers follows.
Males
More
a.
Frequency
0
1
4
18
28
24
14
8
2
1
0
30
25
20
15
10
5
0
1
3
5
7
9
11
13
15
17
19
Midpoint of distance travelled (000's km)
Provide appropriate labels for each of the following:
i.
ii.
the graph title;
the vertical axis.
(3 marks)
b.
Comment on the shape of the distribution.
(2 marks)
c.
Copy the frequency distribution table to your exam booklet.
i.
Complete the table and insert labels for the appropriate class intervals.
(3 marks)
ii.
Add an extra column to your frequency distribution prepared above,
with the heading ‘relative frequency’ and calculate the relative
frequencies for these data.
(3 marks)
QBM117 Exam - Spring 2001
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The frequency distribution generated by MS Excel ® for the distances travelled (in
thousands of kilometres) by female drivers follows.
Females Frequency
2
0
4
4
6
7
8
18
10
24
12
27
14
14
16
6
More
0
d.
Use the statistics functions on your calculator or the formula for grouped data,
to find approximations for the mean and standard deviation number of
kilometres driven by the sample of 100 female drivers.
(4 marks)
e.
Why are the values for the mean and standard deviation determined in d.
above, only approximations?
(2 marks)
The boxplot showing the distribution of the distances travelled (in thousands of
kilometres) by female drivers follows.
BoxPlot
Smallest = 2.3
Q1 = 7.5
Median = 9.85
Q3 = 11.7
Largest = 15.5
IQR = 4.2
Outliers = ()
0.0
5.0
10.0
15.0
20.0
Distance travelled (000's km)
f.
What does the boxplot tell you about the distribution of the distances travelled
by female drivers? Explain
(3 marks)
QBM117 Exam - Spring 2001
Page 3 of 13
Question 2
a.
A batch of invoices being audited has 5% of the invoices in error. A sample
of 10 invoices is taken at random from the batch. What is the probability that
the sample will contain
i.
exactly two incorrect invoices?
(3 marks)
ii.
less than two incorrect invoices?
(2 marks)
iii.
more than 8 correct invoices?
(2 marks)
b.
A Professor of statistics has noted from past experience that students who do
all their assignments and tutorial questions have a 90% chance of passing the
final exam, and if they don’t do any of the assignments and tutorial questions
they have a 15% chance of passing the final exam. The Professor estimates
that 65% of the students do their assignments and tutorial questions.
i.
Define each of the simple events and then draw a probability tree to
represent the information above.
(4 marks)
ii.
What percentage of students passed the final exam?
(2 marks)
iii.
c.
Given that a student passed the final exam, what is the probability they
completed their assignments and tutorial questions?
(2 marks)
The travel time for a truck travelling from Sydney to Brisbane is uniformly
distributed between 14.5 hours and 16 hours.
i.
What is the probability that the trip will take more than 15 hours?
(2 marks)
ii.
What is the probability that the trip will take between 14.5 and 15
hours?
(2 marks)
iii.
What is the probability that the trip will take exactly 15.5 hours?
(1 mark)
QBM117 Exam - Spring 2001
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Question 3
a.
b.
Advertising costs for a 30-second commercial are assumed to be normally
distributed with a mean of $20 000 and standard deviation of $3000.
i.
What is the probability that a given commercial costs between $19 500
and $22 000 to produce?
(4 marks)
ii.
What is the probability that the average cost to produce a sample of
thirty six commercials exceeds $19 500?
(4 marks)
A company surveyed television viewers in an effort to estimate the proportion
of homes with a video cassette recorder (VCR). A survey of 600 homes found
470 with a VCR.
i.
What is the point estimate for the true proportion of homes in the
population with a VCR?
(2 marks)
ii.
What is the 95 percent confidence interval estimate for the proportion
of homes with a VCR?
(4 marks)
iii.
Representatives of the VCR industry claim that the true proportion of
homes with a VCR is 0.80. Based on the confidence interval estimate
in part ii., do the sample data support or refute this claim? Explain.
(2 marks)
iv.
Suppose the company wish to have a sampling error of plus or minus
0.1 in estimating the proportion of homes with a VCR at the 95 percent
confidence level. What size sample is required?
Hint: Use the point estimate in i. when determining the sample size
required.
(4 marks)
QBM117 Exam - Spring 2001
Page 5 of 13
Question 4
a.
A firewood delivery company claims that on average one of their loads weighs
1 ton. Being sceptical of the amount of firewood received a customer
arranged to weigh the next 10 loads delivered to households in his street. The
sample mean was found to be 0.95 tons with a standard deviation of 0.09 tons.
Perform an appropriate statistical test to determine whether or not the
company's claim is correct against the possibility that they actually deliver less
than 1 ton on average. Use a significance level of 0.05. Assume that the
weight of firewood delivered follows a normal distribution.
(8 marks)
b.
The following data and scatterplot represent the 1990 values for fuel economy
(Kilometres/Litre) and lifetime carbon dioxide (CO2) emitted from 12 cars.
Model
Porsche 928
Maserati 222E
Porsche 928
Maserati 222E
Porsche 911
Porsche 911
Porsche 911
Porsche 944
Nissan 240SX
Nissan 240SX
Volkswagon Cabriolet
Volkswagon Cabriolet
Km/l
6.76
6.41
6.76
7.12
7.83
7.83
8.55
9.26
8.90
9.61
9.97
11.39
CO2 (Tons)
61.99
59.52
57.93
56.42
53.62
52.04
49.65
46.23
43.74
42.04
38.54
34.57
65
CO2 [Tons]
60
55
50
45
40
35
30
6.00
8.00
10.00
12.00
Km/L
A simple linear regression analysis was performed using MS Excel ® and the
following outputs generated;
QBM117 Exam - Spring 2001
Page 6 of 13
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.979107
0.958651
0.954517
1.864684
12
ANOVA
df
Regression
Residual
Total
SS
MS
F
Sig. of F
1 806.14142 806.1414 231.8466
0.0000
10 34.770468 3.477047
11 840.91189
Coeffs
Std Error
96.75338 3.1373529
-5.62476 0.3694057
Intercept
Km/l
Frequency
7
6
5
4
3
2
1
0
-4
-2
0
2
4
Residual Midpoint
4
3
Residual
2
1
0
-1 30
40
50
60
70
-2
-3
-4
Predicted CO2 [Tons]
Use the output provided to answer the following questions.
i.
Find the correlation coefficient for the relationship between CO2 emissions
and fuel economy?
(2 marks)
QBM117 Exam - Spring 2001
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ii.
Describe in words the nature of the relationship between CO2 emissions and
fuel economy.
(2 marks)
iii.
Is the relationship between CO2 emissions and fuel economy significant at the
0.05 level? Justify.
(4 marks)
iv.
Predict the lifetime CO2 emissions for a vehicle with a fuel economy of 8
Km/L
(2 marks)
v.
Address the assumption of homoscedasticity for the model.
(2 marks)
QBM117 Exam - Spring 2001
Page 8 of 13
PART B
These questions are to be answered on the General Purpose Answer Sheet provided.
Use a 2B pencil only.
1.
The mean can be calculated for
A.
B.
C.
D.
E.
2.
nominal data
ordinal data
interval data
ratio data
both interval and ratio data
A stem and leaf display which follows, has been used to sort 44 data values.
Scale: 1|1 = 110
1
2
3
4
5
6
7
1
1
0
2
1
1
1
2
2 2 3 5
1 5 5 8 9 9
3 3 3 5 5 5
2 3 4 4 5 6
1 6 6 7 8
1 2 3
5
8
8
8
8
9
Find the median and the mode for these data.
A.
B.
C.
D.
E.
median = 46.5 and mode = 45.
median = 48 and mode = 45.
median = 450 and mode = 450 and 480.
median = 465 and mode = 450.
median = 465 and mode = 450 and 480.
Use the following information to answer questions 3. and 4.
Cars are produced on an assembly line. The number of cars requiring extra work after
assembly measures the quality of these cars. The number of cars with defects (ie those
requiring extra work) for a sample of 12 days follows.
30
34
9
14
28
QBM117 Exam - Spring 2001
9
23
0
5
23
Page 9 of 13
7
0
3.
The average number of defects per day is
A.
B.
C.
D.
E.
4.
The standard deviation number of defects per day is
A.
B.
C.
D.
E.
5.
15.1
15.17
11.5
15.16
16
11.44
11.43
11.94
11.95
8.5
The performance of students from two colleges A and B, are to be compared.
The mean and standard deviation for the marks on a common test are shown
below:
Mean mark
Standard deviation
College A
50
5
College B
70
6
Which of the following statements is correct?
A.
B.
C.
D.
E.
6.
Students from college A performed better on the average than the
students from college B.
All the students from college B performed better than the students
from college A.
All the students from college A performed better than the students
from college B.
The marks of students from college A are relatively more variable than
those of the students from college B.
The marks of students from college B are relatively more variable than
those of the students from college A.
If two events A and B are independent of each other, then P( A | B) equals
A.
B.
C.
D.
E.
0
P( A)
1
P( A and B)
P(B)
QBM117 Exam - Spring 2001
Page 10 of 13
7.
Given that z is the standard normal variable, find P(0.7  z  2.7)
A.
B.
C.
D.
E.
8.
2
0.2385
0.7545
0.7615
0.4772
Given that z is the standard normal variable, find a if P( z  a)  0.7054
A.
B.
C.
D.
E.
0.54
–0.54
0.2611
–0.2611
Can’t be determined because 0.7054 is greater than 0.5.
Use the following information to answer questions 9. and 10.
The average number of cars entering a roundabout, is 5 cars per minute. Cars arrive
randomly and independently.
9.
What is the probability that six or more cars will arrive at the roundabout in
the next minute.
A.
B.
C.
D.
E.
10.
What is the probability that six or more cars will arrive at the roundabout in
the next 3 minutes.
A.
B.
C.
D.
E.
11.
0.762
0.616
0.238
0.384
0.146
0.008
1.152
0.003
0.992
0.997
The normal distribution provides a good approximation to the binomial and
may be used for interval estimation of a population proportion when
A.
B.
C.
D.
E.
n > 30
npˆ and nqˆ are both  5
either npˆ or nqˆ is  5
np and nq are both  5
either np or nq is  5
QBM117 Exam - Spring 2001
Page 11 of 13
12.
A stationery store owner would like to estimate the average retail value of
greeting cards that it has in its inventory. A random sample of 20 greeting
cards indicated a mean value of $2.70 and a standard deviation of $0.35. If
greeting card prices are normally distributed, a 95% confidence interval
estimate of the population mean value of all greeting cards that are in its
inventory would be calculated by
A.
B.
C.
13.
D.
 $0.35 
$2.70  (2.086)

 20 
E.
 $0.35 
$2.70  (2.093)

 20 
A Type II error is made when
A.
B.
C.
D.
E.
14.
the null hypothesis is not rejected when it is false.
the null hypothesis is rejected when it is true.
the alternate hypothesis is accepted when it is false.
the null hypothesis is accepted when it is true.
the alternate hypothesis is accepted when it is true.
A hypothesis test returns a p value of 0.15. The null hypothesis should be
A.
B.
C.
D.
E.
15.
$2.70  2.093  $0.35
 $0.35 
$2.70  (1.96)

 20 
 $0.35 
$2.70  (1.729)

 20 
rejected at the 0.05 level.
rejected at the 0.01 level.
rejected at the 0.10 level.
accepted at the 0.05 level.
none of the above.
A teller at a branch of a savings bank located in a rural community has
averaged 300 transactions daily over the past year. A random sample of 20
days during this year indicates a mean of 295.6 transactions with a standard
deviation of 21.9. The appropriate set of hypotheses to test whether the
population mean daily transactions has decreased is
A.
B.
C.
D.
E.
H 0 :   300
H 0 :   295.6
H 0 :   300
H 0 :   300
H 0 :   300
QBM117 Exam - Spring 2001
H A :   300
H A :   295.6
H A :   300
H A :   300
H A :   300
Page 12 of 13
16.
If x is the mean of a random sample taken from a population which is
normally distributed, the sampling distribution of x
A.
B.
C.
D.
E.
17.
Which of the following values of Pearson's correlation coefficient indicates
the weakest relationship?
A.
B.
C.
D.
E.
18.
correlation analysis.
time series analysis.
regression analysis.
all of the above .
none of the above.
In regression analysis a coefficient of determination of 1 means that
A.
B.
C.
D.
E.
20.
0
0.8
-0.9
0.1
-0.3
If the aim of a study is to predict a continuous random variable y from an
independent continuous random variable x, an appropriate analysis would be
A.
B.
C.
D.
E.
19.
is also normally distributed, provided the sample size is large relative
to the population.
is also normally distributed, provided the sample size is small relative
to the population.
is also normally distributed, regardless of the sample size.
can be highly skewed to the right or the left.
is unable to be determined from the information given.
there is no unexplained variation.
there is a large proportion of unexplained variation.
there is a small proportion of unexplained variation.
as x increases, so does y.
as x decreases, so does y.
The residuals formed when a regression line is fitted to a data set should
ideally
A.
B.
C.
D.
E.
be normally distributed.
have an expected value of zero.
have constant variance.
be independent from each other
possess all the characteristics of A., B., C. and D.
QBM117 Exam - Spring 2001
Page 13 of 13