FACULTY OF SCIENCE AND AGRICULTURE
AUTUMN 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 - Autumn 2001
Page 1 of 13
PART A
These questions are to be answered in the answer booklet provided.
Question 1
The lecturer of business statistics teaching a large lecture wanted to compare the
performance of his students on the three exams that are given during the semester.
The exams each cover one portion of the semester and are not cumulative. The scores
on exam 3, for a sample of 33 students, follow.
Student
1
2
3
4
5
6
7
8
9
10
11
a.
exam 3
74
74
83
71
85
65
88
54
52
62
51
Student
12
13
14
15
16
17
18
19
20
21
22
exam 3
68
48
77
73
64
52
14
25
75
70
84
Student
23
24
25
26
27
28
29
30
31
32
33
exam 3
56
77
48
68
55
61
70
58
96
67
90
Use an ordered stem and leaf display, to sort the scores from exam 3.
(3 marks)
The descriptive statistics for each of the three exams, follows:
exam 1
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
b.
exam 2
74.76
2.40
75
89
13.79
190.13
-0.62
-0.33
57
42
99
2467
33
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
exam 3
67.15
3.48
71
80
19.96
398.51
0.34
-0.78
81
14
95
2216
33
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
3.02
74
1.65
-0.90
14
96
2155
33
Use the display prepared in part a. to help you find the missing values in the
descriptive statistics table for the scores from exam 3. Working does not need
to be shown where the statistics functions on the calculator are used.
(7 marks)
QBM117 Exam - Autumn 2001
Page 2 of 13
The boxplot for each of the three exams, follows:
Boxplot showing results for exam 1
0
20
40
60
80
100
120
Mark as a percentage
Boxplot showing results for exam 2
0
20
40
60
80
100
Mark as a percentage
Boxplot showing results for exam 3
0
20
40
60
80
100
120
Mark as a percentage
Examine all the output provided. Use this output to help you answer the
following questions.
c.
The scores for one exam are more consistent than for the other two exams.
Which exam is this. Give three valid statistical reasons for choosing this exam.
(4 marks)
d.
The lecturer believes he has set a ‘fair’ exam, if at least 75% of students pass
the exam (score more than 50%). Given this condition, which exam(s) would
he consider to have been fair? Explain.
(3 marks)
e.
For which exams would the median provide the best indication of the average
score? Why?
(3 marks)
QBM117 Exam - Autumn 2001
Page 3 of 13
Question 2
a.
b.
c.
A quality control engineer has been asked to examine a complex electronic
system that is currently operating out of control. His examination will involve
testing six switching mechanisms within the system and identifying the one
that is at fault. Assume that one and only one switching mechanism is faulty.
The switching mechanisms are randomly chosen for examination by the
engineer but excluded from further consideration if found to be satisfactory.
i.
What is the probability that the faulty switching mechanism will be
discovered during the second examination? (3 marks)
ii.
What is the probability that the faulty switching mechanism will be
discovered before the fourth examination?
(3 marks)
It is known that 90% of those who purchase a colour television will not have
claims against the guarantee during the duration of the guarantee. Suppose that
each of 25 customers buys a colour television from a certain electrical store.
i.
What is the probability that at least 4 of these 25 customers will have
claims against their guarantee?
(3 marks)
ii.
What is the probability that exactly 5 of these customers will have
claims against their guarantee?
(3 marks)
iii.
What is the expected value and standard deviation of the number of
claims from 25 buyers?
(3 marks)
A manger of a women’s store wishes to determine the relationship between the
type of customer and the type of payment. She has collected the following
data.
Payment
Customer
Regular
Non regular
Credit
70
40
Cash
50
40
If a customer is selected at random, what is the probability that
i.
The customer is regular?
(2 marks)
ii.
The customer is regular or buys on credit?
(3 marks)
QBM117 Exam - Autumn 2001
Page 4 of 13
Question 3
a.
A production process for steel rods, used to reinforce concrete, is known to
produce rods whose lengths have a variance of 64 square centimetres. The
production machinery has been set to produce rods with a mean of 600
centimetres in length. These rods are tied into bundles of 40 for shipment to
the construction sites.
i.
What is the probability that the average length of a randomly selected
bundle is less than 598cm?
(3 marks)
ii.
What is the probability that the average length of a randomly selected
bundle is less than 598cm or more than 601cm?
(3 marks)
iii.
If a construction worker randomly selects one rod from a bundle, what
is the probability that the rod is less than 598cm?
(3 marks)
b.
With increased concern over maintaining adequate cash flow, a builders'
supply company wishes to estimate the average amount owed to it by its credit
customers. For this purpose, 25 of the company's invoices are randomly
sampled and are found to possess an average credit balance of $3200 with a
standard deviation of $350. Calculate a 95% confidence interval estimate for
the average credit balance for all credit customers of the supply company.
(4 marks)
c.
Manufacturers of photographic equipment have introduced many new easy-touse cameras, film types and flash equipment in recent years. A new type of
flashbulb was tested to estimate the proportion of new bulbs that would
produce the required light output at the appropriate time. A sample of 1000
bulbs was tested and 920 were observed to function according to
specifications.
i.
Find a 99% confidence interval estimate for the true proportion of
bulbs functioning according to specifications.
(4 marks)
ii.
How large a sample should be taken in order to estimate the proportion
to within 2%, with 99% confidence?
(3 marks)
QBM117 Exam - Autumn 2001
Page 5 of 13
Question 4
a.
A small soft drink bottle carries a claim on its label that the bottle contains
500ml of soft drink. To examine the validity of this claim, a consumer group
randomly selects a sample of 50 bottles of the soft drink and finds an average
content of 495ml with a standard deviation of 10ml. Test at  = 0.01 to see
whether the data present sufficient evidence to support a claim by the
consumer group that the bottles are under filled.
(8 marks)
b.
The manager of a large company was interested in the effect radio and
television advertising had on sales. He collected data on advertising
expenditure ($000) and sales ($000) for a random sample of 22 weeks. A
simple linear regression analysis of the relationship between advertising
expenditure and sales was performed. The computer output generated by
Excel follows.
Scatterplot showing relationship
between advertising and sales
Sales ($000)
2000
1500
1000
500
0
0
20
40
60
80
Advertising ($000)
SUMMARY OUTPUT
Adv
Regression Statistics
0.79097943
600
R Square
0.62564845
400
Adjusted R Square
0.60693087
Standard Error
216.655953
Observations
Residuals
Multiple R
22
ANOVA
SS
MS
1
1568996.55
1568996.55
Residual
20
938796.041
46939.802
Total
21
2507792.59
Coefficients
Intercept
Advertising
0
-200 0
-400
df
Regression
200
Std Error
Sig F
33.4257172
P-value
-600
1.1735E-05
Lower 95%
Upper 95%
465.086364
139.34134
3.33774861
0.00327931
174.425557
755.747171
16.89
2.92138828
5.78149784
1.1735E-05
10.7960937
22.9839063
QBM117 Exam - Autumn 2001
t Stat
F
Page 6 of 13
2
Advertising Residual Plot
600
Residuals
400
200
0
-200 0
20
40
60
80
-400
-600
Advertising ($000)
Frequency
Histogram of residuals
12
10
8
6
4
2
0
-395.6
-196.0
3.6
203.3
402.9
Residuals
Use the output provided to answer the following questions.
i.
Determine the regression equation to predict sales from advertising
expenditure.
(2 marks)
ii.
One of the required conditions for the validity of regression analysis is
that the error variable must be normally distributed. Explain how we
would verify this from the output provided and comment on whether
this condition has been violated here.
(3 marks)
iii.
From the scatterplot, it appears that sales are linearly related to
advertising expenditure. Test this relationship at a 1% level of
significance.
(4 marks)
iv.
For the estimated model, what is the coefficient of determination.
Interpret this value.
(3 marks)
QBM117 Exam - Autumn 2001
Page 7 of 13
PART B
These questions are to be answered on the General Purpose Answer Sheet provided.
Use a 2B pencil only.
1.
Hungry Jacks has recorded the size of the fries (regular, medium, large, extra
large) ordered by a sample of 100 customers. The data recorded are
A.
B.
C.
D.
E.
2.
You have recorded the heights of five of your classmates. On the basis of this
information, you have made the following statement. "The average height of
all students at the university is 170cm." This is an example of
A.
B.
C.
D.
E.
3.
quantitative and ordinal.
qualitative and ordinal.
quantitative and nominal.
qualitative and nominal.
quantitative and ratio.
a parameter.
a population
descriptive statistics
inferential statistics.
a distribution.
The salaries in thousands of dollars of 50 CEO’s from the top corporations is
given in the following frequency table.
Salary (in $000's)
>90 up to and including 440
>440 up to and including 790
>790 up to and including 1140
>1140 up to and including 1490
>1490 up to and including 1840
>1840 up to and including 2190
>2190 up to and including 2540
Frequency
9
11
10
8
4
3
5
Excel has been used to construct a histogram to represent the frequency
distribution above. The most correct histogram would be
QBM117 Exam - Autumn 2001
Page 8 of 13
A.
-1
14
>1
0
14
0
-1
49
>1
0
49
0
-1
84
>1
0
84
0
-2
19
>2
0
19
0
-2
54
0
>7
90
>4
40
>9
0
-7
90
12
10
8
6
4
2
0
-4
40
Frequency
Salaries of 50 of the top CEO's
Salaries ($1000's)
B.
Frequency
Salaries of 50 of the top CEO's
15
10
5
0
440
790 1140 1490 1840 2190 2540
Salaries ($1000's)
C.
Frequency
Salaries of 50 of the top CEO's
15
10
5
0
440
790 1140 1490 1840 2190 2540
Salaries ($1000's)
QBM117 Exam - Autumn 2001
Page 9 of 13
D.
Frequency
Salaries of 50 of the top CEO's
15
10
5
0
265
615
965 1315 1665 2015 2365
Salaries ($1000's)
E.
Frequency
Salaries of 50 of the top CEO's
15
10
5
0
265
615
965 1315 1665 2015 2365
Salaries ($1000's)
4.
Which of the following statements is true?
A.
B.
C.
D.
E.
5.
The best type of chart for comparing two sets of categorical data is
A.
B.
C.
D.
E.
6.
When the distribution is skewed to the left, mean > median > mode.
When the distribution is skewed to the right, mean < median < mode.
When the distribution is skewed to the left, mean > median.
When the distribution is unimodal and symmetric, mean = median =
mode.
When the distribution is bimodal and symmetric, mean = median =
mode.
a line chart.
a frequency distribution.
a histogram.
a frequency polygon.
a grouped bar chart.
Given that z is a standard normal random variable, find P( z  1.8)
A.
B.
C.
D.
E.
0.9641
0.8599
0.4641
0.3599
0.0359
QBM117 Exam - Autumn 2001
Page 10 of 13
7.
If A and B are mutually exclusive events where P(A) = 0.2 and P(B) = 0.3,
then P(A and B) is
A.
B.
C.
D.
E.
0.05
0.6
0.06
0
0.1
Use the following information to answer questions 8. and 9.
The number of computer malfunctions in an accountancy firm occur randomly and
independently at an average rate of 3 malfunctions per month.
8.
Calculate the probability that at least three malfunction will occur in a given
month.
A.
B.
C.
D.
E.
9.
Calculate the probability that less than four malfunctions will occur in the next
two months.
A.
B.
C.
D.
E.
10.
0.647
0.285
0.151
0.715
0.849
Given that z is a standard normal random variable, find the value for c such
that P(c  z  c)  0.8664
A.
B.
C.
D.
E.
11.
0.353
0.577
0.224
0.423
0.647
1.5
0.4332
0.1664
-1.5
cannot be determined from the information provided.
If P(A|B) = 0.6, P(A) = 0.5 and P(B) = 0.4, then P(A or B) is
A.
B.
C.
D.
E.
0.66
0.90
0.24
0.16
0.50
QBM117 Exam - Autumn 2001
Page 11 of 13
12.
A home owner claims that the current market value of his house is
$250 000. An assessor believed the house to be worth more than $250 000.
Sixty real estate agents were asked independently to estimate the house’s
value. The hypothesis test that followed ended with a decision to ‘reject H 0 ’.
Which of the following statements accurately states the conclusion?
A.
B.
C.
D.
E.
13.
The home owner is correct, the house is worth $250 000.
The home owner is correct, the house is worth more than $250 000.
The home owner is incorrect, the house is worth less than $250 000.
The home owner is incorrect, the house is worth more than $250 000.
The home owner is incorrect, he should not sell his house.
A given sample for a hypothesis test yields a p-value of 0.02. For this situation
A.
at  = 0.001 we reject H 0 .
B.
C.
at  = 0.01 we reject H 0 .
at  = 0.05 we reject H 0 .
D.
E.
at  = 0.1 we do not reject H 0 .
rejection of H 0 depends on whether we have a 1-tailed or 2-tailed test.
Use the following information to answer questions 14 and 15.
We are performing a hypothesis test where H 0 :   50 , H A :   50 ,
x  53.2 s  20 n  49   0.05 .
14.
The test statistic would be
A.
B.
C.
D.
E.
15.
z = 1.12
z = -1.12
t = 1.12
t = -1.12
t = 0.16
The decision rule would be
A.
reject H 0 if t sample  1.676
B.
reject H 0 if t sample  2.009
C.
reject H 0 if z sample  1.645
D.
reject H 0 if z sample  1.96
E.
reject H 0 if t sample  2.009
QBM117 Exam - Autumn 2001
Page 12 of 13
16.
When performing a simple linear regression, the smallest value the standard
error of estimate can take is
A.
B.
C.
D.
E.
17.
Which of the following statistics and procedures can be used to determine
whether a linear regression model should be employed.
A.
B.
C.
D.
E.
18.
there is no explained variation.
there is no unexplained variation.
there is no y-intercept in the model.
there are outliers.
the two variables are not linearly related.
Given the least squares regression line y = 5 - 2x
A.
B.
C.
D.
E.
20.
the standard error of estimate
the coefficient of determination
the t - test of the slope
the residual plot.
all of the above.
The Pearson correlation coefficient r is 1 when,
A.
B.
C.
D.
E.
19.
2
-2
1
-1
0
The y-intercept of the regression line is -2
The relationship between x and y is positive.
The relationship between x and y is negative.
As x increases, so does y.
As x decreases, so does y.
Which value of the correlation coefficient r, indicates a stronger correlation
than 0.65?
A.
B.
C.
D.
E.
0.45
0.6
-0.75
0.7
both 0.7 and -0.75.
QBM117 Exam - Autumn 2001
Page 13 of 13