FACULTY OF SCIENCE AND AGRICULTURE
AUTUMN SESSION EXAMINATION 1999
QBM 117 BUSINESS STATISTICS
SUBJECT CONVENOR:
Kerrie Cullis (Wagga Wagga)
DAY & DATE:
TIME:
WRITING TIME:
Three (3) hours
READING
minutes
TIME:
Ten (10)
MATERIALS SUPPLIED BY UNIVERSITY:
1 x 24 page examination answer
booklet
1 x General Purpose Answer
Sheet
MATERIALS PERMITTED IN EXAMINATION:
Battery operated calculator (no printer)
2B Pencil/eraser
Text: Australian Business Statistics
Selvanathan and Selvanathan OR
Stats for Business and Economics
Anderson et al
May
be
annotated
and
cross
referenced.
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 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 booklet provided. Number each question
clearly. Write your name and student number on the front cover of the answer booklet 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 Examination - Autumn 1999
Page 1 of 16
PART A
These questions are to be answered in the answer booklet provided.
Question 1
In many manufacturing processes there is a term called work in progress (often
abbreviated WIP). In a book manufacturing plant this represents the time it takes for
sheets from a press to be folded, gathered, sewn, tipped, and bound. The following
data represent samples of 20 books at each of two production plants and the
processing time (operationally defined as the time in days from when the books came
off the press to when they were packed in cartons) for these jobs.
4.42
8.58
9.54
5.75
a.
5.29
8.92
11.46
12.46
5.41
9.29
16.62
9.17
5.62
10.50
Plant A
7.29
7.50
10.92 11.42
7.54
11.46
7.58
11.62
7.96
16.25
8.45
21.62
12.62
13.21
Plant B
25.75 15.41
6.00
2.33
14.29
14.25
13.13
5.37
13.71
6.25
10.01
9.71
Find the mean and standard deviation processing times for Plant A.
(4 marks)
b.
Determine the median and mode processing times for Plant A.
(3 marks)
Use your answers from a. – b. above together with the output following to answer
question c. and d.
Descriptive Statistics
Plant B
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
QBM117 Examination - Autumn 1999
11.35
1.15
11.96
#N/A
5.13
26.28
2.18
0.80
23.42
2.33
25.75
227.07
20
Page 2 of 16
c.
Are the processing times at Plant A and Plant B skewed? If so, in which
direction? Explain how you reached your conclusions by referring to your
answers in a. and b. above and the MS Excel output provided.
(5 marks)
d.
On the basis of the results in a. – c. above and the MS Excel output provided,
are there any differences in the processing times at the two plants. Explain.
(4 marks)
e.
In the Descriptive Statistics table for Plant B, explain the meaning of #N/A for
the mode.
(2 marks)
f.
In the Descriptive Statistics table for Plant B, how is the value for the standard
error obtained?
(2 marks)
QBM117 Examination - Autumn 1999
Page 3 of 16
Question 2
a.
A survey of a magazine’s subscribers indicates that 60% own a home and 75%
own a car. Ninety percent of the home owners who subscribe to the magazine,
also own a car. What proportion of subscribers
i.
own both a car and a house?
(3 marks)
ii.
own a car or a house?
(2 marks)
iii.
own neither a car nor a house?
(2 marks)
b.
c.
A certain brand of flood lamps has a length of life that is normally distributed
with a mean of 3500 hours and a standard deviation of 200 hours.
i.
What proportion of these lamps will last for more than 4000 hours?
(3 marks)
ii.
What length of life should the manufacturers advertise for these lamps
in order that only 3% of the lamps will burn out before the advertised
length of life?
(3marks)
An advertisement claims that two out of five doctors recommend a certain
pharmaceutical product. A random sample of 20 doctors is selected, and it is
fond that only two of them recommend the product.
i.
Assuming the advertising claim is true, what is the probability of the
observed event ie only two doctors recommending the product?
(3 marks)
ii.
Assuming the claim is true, what is the probability of two or fewer
doctors recommending the product?
(2 marks)
iii.
Given the results in i. and ii. above, do you believe the advertisement?
Explain.
(2 marks)
QBM117 Examination - Autumn 1999
Page 4 of 16
Question 3
a.
b.
A survey of 100 retailers revealed that the mean after-tax profit was $75 000.
We assume that the population standard deviation is $20 000,
i.
Find the 99% confidence interval estimate of the mean after-tax profit
for all retailers.
(5 marks)
ii.
If we wish to estimate the mean after-tax profit with 99% confidence
and to within $4 000, how many additional retailers would need to be
surveyed?
(5 marks)
A statistician hires a company to input survey data onto a computer. The
company claims that the error rate is less than 0.1%. To test the claim the
statistician examines 10 000 numbers and discovers that three are incorrect.
Do these results support the company’s claim? Test with  = 0.01.
(10 marks)
QBM117 Examination - Autumn 1999
Page 5 of 16
Question 4
a.
A brokerage would like to be able to predict the number of trade executions
per day and has decided to use the number of incoming phone calls as a
predictor variable. Data were collected over a period of 35 days. The data were
plotted using MS Excel and the scatterplot follows.
Trade Executions
Scatter plot of no. incoming calls vs trade
executions
500
450
400
350
300
250
200
1700
1900
2100
2300
2500
2700
No. incoming calls
i.
Comment on the apparent relationship between trade executions and
the number of incoming phone calls as evident from the scatterplot.
(2 marks)
MS Excel has been used to fit a simple linear regression. The output produced
by MS Excel follows.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.79377053
R Square
0.63007166
Adjusted R Square
0.61886171
Standard Error
29.4190391
Observations
35
ANOVA
df
Regression
Residual
Total
Intercept
Calls
SS
MS
F
1 48645.56 48645.6 56.2065
33 28560.84 865.48
34
77206.4
Coefficients Std Error
-63.0204576 54.59737
0.18900568 0.025211
QBM117 Examination - Autumn 1999
Sig. F
1.28E-08
t Stat
P-value Lower 95%
-1.1543 0.25668 -174.0997
7.4971 1.3E-08 0.137714
Page 6 of 16
Residuals
Calls Residual Plot
80
60
40
20
0
-201700
-40
-60
-80
1900
2100
2300
2500
2700
Calls
ii.
State the regression equation, expressing each coefficient to 2 decimal
places
(2 marks)
iii.
Interpret the meaning of the slope coefficient in this problem.
(2 marks)
iv.
Use the regression equation, stated in ii. above to predict the average
number of trades executed for a day in which the number of incoming
calls is 2000.
(1mark)
v.
Would it be appropriate to use the model to predict the average number
of trades executed for a day in which the number of incoming calls is
5000? Explain.
(2 marks)
Given x  2156.66 and SS x  1 361 738 for the number of incoming
calls, calculate a 95% confidence interval estimate of the average
number of trades executed for days in which the number of incoming
calls is 2000.
(3 marks)
The following data represent the annual number of employees (in thousands)
for an oil supply company for the years 1978 – 1997.
vi.
b.
Number of employees (in thousands)
Year
Number
Year
Number
1978
1.45
1985
2.04
1979
1.55
1986
2.06
1980
1.61
1987
1.80
1981
1.60
1988
1.73
1982
1.74
1989
1.77
1983
1.92
1990
1.90
1984
1.95
1991
1.82
QBM117 Examination - Autumn 1999
Year
1992
1993
1994
1995
1996
1997
Number
1.65
1.73
1.88
2.00
2.08
1.88
Page 7 of 16
i.
If the data is smoothed using a 3-year moving average, find these
moving averages for 1993 - 1996.
(4 marks)
ii.
Can we calculate a 3-year moving average for 1997? Explain.
(2 marks)
iii.
The original data together with the smoothed data have been plotted in
the graph below. The graph of the moving averages is incomplete. Plot
the 3-year moving averages for 1993 – 1996 calculated in i. above, on
the graph below.
(2 marks)
Annual no. employees in an oil supply company
2.2
No. employees (000's)
2.1
2
1.9
1.8
1.7
1.6
1.5
Year
QBM117 Examination - Autumn 1999
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1.4
No. Employees
3yr MA
Page 8 of 16
PART B
These questions are to be answered on the General Purpose Answer Sheet provided.
Use a 2B pencil only.
1.
A researcher has collected the following sample data. The mean of the sample
is 5.
3
5
12
3
2
The coefficient of variation is
A.
B.
C.
D.
E.
2.
3.
Information was obtained from students as they left the University Coop
bookshop during the first week of classes. Which of the following variables
would be classified as ratio data.
A.
B.
C.
D.
The course being studied.
The type of computer owned by the student
The amount of money spent on books.
The method of payment.
E.
The students progress through the course ie 1st year, 2nd year, etc.
Which of the following measures of spread (dispersion) is least affected by the
presence of extreme (outlying) values?
A.
B.
C.
D.
E.
4.
0.727
0.812
2.640
3.300
0.264
Variance
Range
Median
Interquartile range
Standard deviation
The lifetimes of a sample of 40, 100-watt light globes were organised into the
following frequency distribution using MS Excel.
Bins
Frequency
700
800
900
1000
1100
More
QBM117 Examination - Autumn 1999
2
2
13
16
7
0
Page 9 of 16
The histogram which most correctly represents this data is given by
A.
Histogram of lifetime of a sample of 40
100-watt light globes
Frequency
20
15
10
5
0
700
800
900
1000
1100 More
Lifetim e (hours)
B.
Histogram of lifetime of a sample of 40
100-watt light globes
Frequency
20
15
10
5
M
or
e
-1
20
0
-1
10
0
11
00
-1
00
0
10
00
-9
00
90
0
80
0
70
0
-8
00
0
Lifetim e (hours)
C.
Histogram of lifetime of a sample of 40
100-watt light globes
Frequency
20
15
10
5
M
or
e
-1
10
0
10
00
-1
00
0
90
0
-9
00
80
0
-8
00
70
0
60
0
-7
00
0
Lifetim e (hours)
QBM117 Examination - Autumn 1999
Page 10 of 16
D.
Histogram of lifetime of a sample of 40
100-watt light globes
Frequency
20
15
10
5
0
650
750
850
950
1050
Lifetim e (hours)
E.
Histogram of lifetime of a sample of 40
100-watt light globes
Frequency
20
15
10
5
0
750
850
950
1050
1150
Lifetim e (hours)
5.
The reason why the standard deviation of a distribution is more easily
interpreted than the variance is that
A.
B.
C.
D.
E.
it is a measure of average deviation.
it is expressed in the same units as the data.
it takes every data value into account.
it is standardised.
it takes the mean into account.
QBM117 Examination - Autumn 1999
Page 11 of 16
Use the following information to answer questions 6. and 7.
An advertising executive receives an average of 10 phone calls each afternoon
between 2pm and 4pm. The calls occur randomly and independently of one another.
6.
Find the probability that the executive will receive at least three calls between
2pm and 4pm on a particular afternoon.
A.
B.
C.
D.
E.
7.
Find the probability that the executive will receive 12 calls between 2pm and
3pm on a particular afternoon.
A.
B.
C.
D.
E.
8.
0.9671
0.5329
0.4671
0.0500
0.0329
If P(A) = 0.4, P(B) = 0.6 and P(B|A) = 0.75 then P(A  B) is
A.
B.
C.
D.
E.
10.
0.792
0.095
0.998
0.003
0.995
If Z is a standard normal random variable, find P( Z  1.84)
A.
B.
C.
D.
E.
9.
0.997
0.003
0.010
0.007
0.990
0.45
0.53
0.80
0.24
0.30
Consider the process of tossing a fair coin 6 times. Which of the following
sequences of tosses is more likely to occur?
A.
B.
C.
D.
E.
THTHHT
HHHTTT
HHHHHH
TTHHTH
They are all equally likely.
QBM117 Examination - Autumn 1999
Page 12 of 16
11.
The amount of time a person is required to wait for an elevator in a large
department store is uniformly distributed with a range of waiting time from no
wait (0 seconds) to 3 minutes and 30 seconds wait. A recent study in consumer
behaviour suggests that if customers are required to wait more than two
minutes and 45 seconds, they develop negative impressions of the commitment
of the store to customer service. Based on this study, what proportion of
customers using the department store’s elevator are likely to develop negative
impressions of the store’s commitment to customer service?
A.
B.
C.
D.
E.
12.
A population has a mean of 53 and a standard deviation of 21. A sample of 49
observations will be taken. The probability that the sample mean will be less
than 57.95 is:
A.
B.
C.
D.
E.
13.
14.
31.25%
68.75%
21.43%
25.76%
78.57%
0.0948
0.0495
0.4505
0.9505
0.5948
A random sample of 25 students were found to have an average age of 25 years
with a standard deviation of two years. The 90% confidence interval for the
true age of students is calculated using
A.
x  t 25, 0.1 (0.4)
B.
x  t 25, 0.05 (0.4)
C.
x  t 24, 0.1 (0.4)
D.
x  t 24, 0.05 (0.4)
E.
x  Z 0.05 (0.4)
A human resources manager stated that after sampling he could report the
standard error of response times to service calls to be 20 minutes. Given that
the variance of response times to calls is known to be 40 000 minutes, find the
size of the sample taken.
A.
B.
C.
D.
E.
10
100
25
2000
4 000 000
QBM117 Examination - Autumn 1999
Page 13 of 16
15.
Given the same sample mean and variance, which of the following would give
the widest confidence interval for the population mean?
A.
B.
C.
D.
E.
16.
The residuals formed when a regression line is fitted to a data set should
ideally
A.
B.
C.
D.
E.
17.
A 95% confidence interval with n = 36.
A 95% confidence interval with n = 100.
A 99% confidence interval with n = 36.
A 99% confidence interval with n = 100.
All the above intervals would have the same width.
be normally distributed.
have an expected value of zero.
have a variance which is independent of the value of the independent
variable.
be independent from each other.
possess all the characteristics of A., B., C. and D.
A car rental agency wants to determine the correlation between monthly
maintenance costs and the distance travelled by their smaller cars. Data on 15
randomly selected cars were plotted in the graph following.
Monthly maintenance cost vs
distance travelled
y = 1.8055x + 8.6546
R2 = 0.9351
100
Cost ($)
80
60
40
20
0
0
10
20
30
40
Distance travelled (000's km)
From the information provided in the graph, the sample correlation coefficient
measuring the correlation between maintenance costs and distance travelled is
A.
B.
C.
D.
E.
0.97
0.87
0.94
-0.97
unable to be determined from the information provided.
QBM117 Examination - Autumn 1999
Page 14 of 16
18.
Consider the following set of seasonal-irregular component values for a
quarterly time series.
Quarter 1
1.102
1.064
1.139
Quarter 2
0.938
1.007
0.979
Quarter 3
0.933
0.901
0.905
Quarter 4
1.009
1.075
0.950
The seasonal index for the third quarter would be
A.
B.
C.
D.
E.
19.
0.993
0.996
2.739
1.000
0.913
The following seasonal indices and trend line were computed from five years
of quarterly sales data. Forecast the sales for the second quarter of the sixth
year.
yˆ  500  30t
(t  1, 2, 3, ... ,20)
Quarter
1
2
3
4
A.
B.
C.
D.
E.
Seasonal index
1.4
1.2
0.9
0.5
1130
1356
1392
1160
1582
QBM117 Examination - Autumn 1999
Page 15 of 16
20.
The data in the following table represent weekly sales (in thousands) for
computer disks. Exponentially smoothed sales data (w = 0.4) are also shown
however the value for week 5 is missing.
Week
1
2
3
4
5
Sales (000’s)
57
58
60
54
56
Sales (000’s) (w = 0.4)
57.0
57.4
58.4
56.7
The exponentially smoothed value for week 5 would be
A.
B.
C.
D.
E.
56.0
56.4
55.6
56.3
56.7
QBM117 Examination - Autumn 1999
Page 16 of 16