FACULTY OF SCIENCE AND AGRICULTURE
SPRING SESSION EXAMINATION 1999
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 24 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 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 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 Examination - Spring 1999
Page 1 of 14
PART A
These questions are to be answered in the answer booklet provided.
Question 1
A Pizza Shop monitored (over a two week period) the time taken from
placement of order until the pizza was delivered. The data were summarized
and the following ogive generated.
Cumulative Relative Frequency
a.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
i.
5 10 15 20 25 30 35 40 45 50 55 60 65 70
Delivery Time (minutes)
Write an appropriate title for this graph
(1 mark)
ii.
What proportion of pizzas were delivered within 30 minutes ?
(1 mark)
iii.
If there were 2500 pizzas delivered during the sampling period, calculate how
many pizzas took between 15 and 45 minutes to deliver.
(2 marks)
iv.
The Pizza Shop has decided to offer a refund if pizzas aren't delivered within a
certain time period. They calculate that they can afford this refund for 5
percent of their orders. What delivery time should they set as the cut off for
this offer?
(1 mark)
QBM117 Examination - Spring 1999
Page 2 of 14
b.
A manufacturer is developing a new nickel-metal hydride battery to be used
in mobile phones as a possible replacement for the old nickel-cadmium
batteries. The director of quality control decides to evaluate the newly
developed battery against the widely used old battery with respect to
performance. A random sample of 25 of the old batteries, and 25 of the new
batteries are placed in mobile phones of the same brand and model. The
performance measure of interest is the talking time (in minutes) prior to
recharging. The results are as follows. The times listed are in numerical order,
rather than the order in which they were recorded.
Old battery
Talking time (in minutes)
220.8 221.7 234.5 236.7 238.7 244.5 244.9 247
247.8 248.8
249.7 250.4 251
251.8 252.5 252.8 254.9 255.4 256.9 261
262
263.3 266.8 270.4 284.4
New battery
Talking time (in minutes)
221.1 226.4 242.5 249.4 251.8 252.1 254.1 254.3 257.5 258.3
259.8 261.3 262.3 262.8 263.2 265
265.3 265.3 265.5 266.1
267.3 271.1 275.4 283
292.3
i.
Calculate the mean and standard deviation talking time for the old
battery. (Use the statistics functions on your calculator to find these
values).
(2 marks)
The following table shows the talking time descriptive statistics, for the newly
developed battery.
New battery
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
ii.
259.728
3.006856
262.3
265.3
15.03428
226.0296
1.900702
-0.64261
71.2
221.1
292.3
6493.2
25
Are the times for each type of battery equally variable? Use values
from the descriptive statistics table, together with any other necessary
calculations to justify your answer.
(3 marks)
QBM117 Examination - Spring 1999
Page 3 of 14
The following boxplot can also be used to compare the talking times for the
two batteries.
iii.
Are the talking times for the old battery skewed? If so, include the
direction of skewness and justification for your conclusion.
(3 marks)
iv.
Which measure of centre is most useful, and which is least useful in
describing the average talking time on a mobile phone in which the
newly developed battery is used. Explain.
(4 marks)
v.
From the information provided here, does one battery appear to
perform better? Explain.
(3 marks)
QBM117 Examination - Spring 1999
Page 4 of 14
Question 2
a.
An import-export firm has a 45% chance of concluding a deal to export
agricultural equipment to a developing nation, if a major competitor does not
bid for the contract, and a 25% probability of concluding the deal if the
competitor does bid for it. It is estimated there is a 40% chance that the
competitor will submit a bid for the contract.
i.
Define each of the simple events and then prepare a probability tree to
represent this problem. Ensure the relevant probabilities appear along
each branch of the tree.
(3 marks)
ii.
What is the probability of concluding the deal?
(3 marks)
iii.
b.
If the firm secures the deal, what is the probability that the competitor
had also put in a bid?
(2 marks)
A Perth interviewer has found that one in five of the people approached will
refuse to take part in her survey. Of the next 20 people approached, what is the
probability that
i.
exactly five will agree to an interview?
(3 marks)
ii.
at least half will agree to an interview?
(3 marks)
c.
Beer sales at a Darwin drive-in bottle shop are normally distributed with an
average of 3000 litres per week, and a variance of 40 000.
i.
Determine the probability that in any particular week, sales will exceed
3200 litres.
(3 marks)
ii.
Determine the beer sales level that is exceeded for 95% of the weeks.
(3 marks)
QBM117 Examination - Spring 1999
Page 5 of 14
Question 3
a.
The subject coordinator for QBM117 has determined the average grade for all
final exams over the last ten years to be 70.3, with a standard deviation of 9.3.
If the Spring ’99 class of 180 students studying QBM117 is considered as a
sample, find the probability that the class's average score on the final exam
will be more than 71.
(4 marks)
b.
It is claimed on the packaging of MicroPop microwave popcorn that only 2
per cent of its kernels fail to pop. A statistics student believes the percentage
to be larger. She decides to test the claim (at the 0.01 significance level). She
selects a sample of 2000 kernels and finds that 44 fail to pop.
i.
State the hypothesis test.
(2 marks)
ii.
Determine whether the test results support the packaging claim.
(7 marks)
iii.
Find the p-value for the test.
(2 marks)
iv.
c.
Explain how the p-value calculated in iii. can be used to test the
hypotheses stated in i.
(2 marks)
How many prawns should be sampled from a catch if we want to be 95%
confident, that the sample mean is no more than 1.4 grams from the actual
mean weight of the catch? Assume that the range of weights in the catch is
about 17 grams.
(3 marks)
QBM117 Examination - Spring 1999
Page 6 of 14
Question 4
a.
A computer manufacturer is considering the purchase of a bulk lot of
microprocessor chips that are rated at a speed of 233 megahertz (MHz), but
the rating is conservative and usually the chips run at a higher speed. Eight
chips are randomly selected and tested, and found to have the following
speeds in megahertz.
234.3 233.6 232.9 235.2 234.0 232.8 233.8 233.0
Construct a 90% confidence interval to estimate the mean operating speed of
the chips in the bulk lot. Assume that the operating speeds are approximately
normally distributed.
(5 marks)
b.
The management of one of the major building societies would like to know
whether there is a relationship between the number of people employed by the
building societies and the number of building society members (clients). Data
on the number of employees and the number of members was collected for a
random sample of 48 major building societies in Australia. Simple linear
regression analysis was performed on these data and the MS Excel output is
provided below.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.90400774
R Square
0.81722998
Adjusted R Square
0.81325672
Standard Error
8668.33322
Observations
48
ANOVA
df
Regression
Residual
Total
Intercept
No. employees
1
46
47
SS
MS
F
Significance F
15454977350 1.545E+10 205.68242 1.36095E-18
3456440041 75140001
18911417391
Coefficients Standard Error
t Stat
P-value
Lower 95%
Upper 95%
8369.25942
2296.040222 3.645084 0.000678 3747.574578 12990.9443
341.133217
23.78621945 14.341632 1.361E-18 293.2540863 389.012348
QBM117 Examination - Spring 1999
Page 7 of 14
No. members
Scatterplot of the number of employees and the
number of members for a sample of 48 of the major
Australian credit unions
150,000
100,000
50,000
0
0
100
200
300
400
No. employees
No. employees Residual Plot
Residuals
40000
20000
0
-20000 0
100
200
300
400
-40000
No. employees
Histogram of residuals
Frequency
20
15
10
5
-1
70
00
-1
04
00
-3
80
0
28
00
94
00
16
00
0
22
60
0
29
20
0
0
Residuals
QBM117 Examination - Spring 1999
Page 8 of 14
Use the MS Excel output provided to answer the following questions.
i.
From the scatterplot provided, comment on the likely relationship
between the number of employees and the number of members.
(2 marks)
ii.
There is an obvious outlier in the data. If this was removed, because it
was suspected it was a data entry error, what impact would this have
on the relationship proposed in part i.? Explain
(2 marks)
iii.
Assuming the relationship proposed in i., write down the sample
regression equation.
(2 marks)
iv.
Determine whether the number of employees influences the number of
members. An appropriate hypothesis test should form part of this
analysis. [Use   0.05 ]
(3 marks)
v.
Which of the diagnostic plots provided would we examine if we
wanted to check whether the fitted model was appropriate for the credit
union data? Explain whether the fitted model is appropriate here and
how you reached this conclusion.
(2 marks)
vi.
Find the 95% confidence interval for the expected number of members
of credit unions with 220 employees.
Use SS x  132806.8 and x  80.935
(4 marks)
QBM117 Examination - Spring 1999
Page 9 of 14
PART B
These questions are to be answered on the General Purpose Answers Sheet
provided. Use a 2B pencil ONLY.
1.
Which of the following statements is true?
A.
B.
C.
D.
E.
2.
Which of the following is best for comparing the dispersion of two data sets
with different means?
A.
B.
C.
D.
E.
3.
The standard deviation complements the mean when describing data.
The standard deviation must take a positive number.
The standard deviation increases as the spread of the data increases.
The standard deviation is a good descriptor of skewed data
The standard deviation is the square root of the variance.
Find the median of the following data: {2, 4, 8, 9, 19, 6}.
A.
B.
C.
D.
E.
5.
Range
Variance
Standard deviation
Boxplots
Coefficient of variation
Which of the following is false?
A.
B.
C.
D.
E.
4.
The median must be one of the values of the raw data.
The mean must be one of the values of the raw data.
The mode does not have to be one of the values of the raw data.
The standard deviation must be one of the values of the raw data.
The first quartile does not have to be one of the values of the raw data.
7
6
8.5
8
17
Which of the following is cannot be determined from the boxplot?
A.
B.
C.
D.
E.
The median.
The range.
The first quartile.
The mean.
A., B., C., and D. can all be determined from the boxplot.
QBM117 Examination - Spring 1999
Page 10 of 14
6.
The figure following displays a density curve with three points marked. Which
of the following statements about these three points is correct?
A B C
A.
B.
C.
D.
E.
A represents the mean, B the median and C the mode.
A represents the median, B the mode and C the mean.
A represents the mode, B the median and C the mean.
A represents the mean, B the mode and C the median.
A represents the median, B the mean and C the mode.
Use the following information to answer questions 7. and 8.
25% of the population in a given area is exposed to a television commercial for Ford
automobiles, and 34% is exposed to Ford’s radio advertisements. Also, it is known
that 10% of the population is exposed to both means of advertising.
7.
If a person is selected at random, from the population in this area, what is the
probability that he or she was exposed to at least one of the two modes of
advertising?
A.
B.
C.
D.
E.
8.
0.59
0.1
0.49
0.51
0.69
The events exposed to a television commercial for Ford and exposed to a radio
commercial for Ford are
A.
B.
C.
D.
E.
mutually exclusive events.
independent events.
dependent events.
disjoint events.
complementary events.
QBM117 Examination - Spring 1999
Page 11 of 14
Use the following information to answer questions 9. and 10.
Many Telstra customers now choose to pay their telephone bill by phone, using a
credit card. Over the past year, Telstra have found that on average, 10 customers per
minute, phone in to pay their bill. If we assume these customers phone in randomly
and independently,
9.
What is the probability of fewer than 7 customers phoning to pay their bill in
the next minute?
A.
B.
C.
D.
E.
10.
0.220
0.090
0.012
0.130
0.063
What is the probability of exactly 8 customers ringing to pay their bill in the
next 30 seconds?
A.
B.
C.
D.
E.
0.333
0.065
0.932
0.113
0.036
Use the following information to answer questions 11. and 12.
The number of orders for installation of a computer information system arriving at an
agency per week is a random variable X with the following probability distribution.
X
P(X)
11.
0
0.10
2
0.30
3
0.15
4
0.15
5
0.05
What is the probability that either 4 or five orders will arrive in a given week?
A.
B.
C.
D.
E.
12.
1
0.20
0.15
0.05
0.0075
0.20
0.85
Assuming independence of weekly orders, what is the probability that three
orders will arrive next week and the same number of orders the following
week?
A.
B.
C.
D.
E.
0.2025
0.0225
0.45
0.30
0.15
QBM117 Examination - Spring 1999
Page 12 of 14
6
0.05
13.
Given Z is the standard normal random variable, find z0 such that
P( Z  z0 )  0.12
A.
B.
C.
D.
E.
14.
0.9785
0.9759
0.0215
0.4785
0.0013
needs more information before reaching a conclusion.
should reject the null hypothesis.
should increase the level of significance.
should not reject the null hypothesis.
should reject the p-value.
Samples are drawn from a population and the sample means calculated. The
resulting sampling distribution of the sample mean
A.
B.
C.
D.
E.
17.
z0  1.175
When performing a hypothesis test at   0.05 , the p-value is calculated as
0.0411. The statistician
A.
B.
C.
D.
E.
16.
z0  1.175
z0  0.30
Given Z is the standard normal random variable, find P(2  Z  3)
A.
B.
C.
D.
E.
15.
z0  0.0478
z0  0.38
has a variance that is smaller than the population variance.
has an expected value that increases with sample size.
is always normal.
always has the same shape as the population.
has a variance that will increase with increasing sample size.
A political party wants to investigate whether the proportion of people voting
for it has changed from the previous proportion of 0.5. 110 people from a
random sample of 200 indicated they would vote for the party. The
appropriate hypotheses to test this would be:
A.
B.
C.
D.
E.
H o : p  0.5 H A : p  0.5
H o : p  0.5 H A : p  0.5
H o : p  0.55 H A : p  0.55
H o : p  0.55 H A : p  0.55
H o : p  0.55 H A : p  0.55
QBM117 Examination - Spring 1999
Page 13 of 14
18.
Given the same sample proportion of 0.45, which of the following would lead
to the widest confidence interval for estimating the population proportion?
A.
B.
C.
D.
E.
A 95 percent confidence interval with n=100
A 95 percent confidence interval with n=50
A 99 percent confidence interval with n=100
A 99 percent confidence interval with n=50
A 90 percent confidence interval with n=100
Use the following information to answer question 19. and 20.
A company that has the distribution rights to home video sales of previously released
movies would like to estimate the correlation between box office gross and the
number of videos sold. The data for a sample of 30 movies together with their box
office gross (in millions of dollars) and the number of videos sold (in thousands) were
analysed using a simple linear regression. The output appears below.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.85308869
R Square
0.72776032
Adjusted R Square
0.71803748
Standard Error
47.8667885
Observations
30
ANOVA
df
Regression
Residual
Total
Intercept
Box office gross
19.
SS
MS
F
Significance F
171499.778 171499.78 74.850547
2.1259E-09
64154.42435 2291.2294
235654.2023
Coefficients Standard Error
t Stat
76.5351424
11.83184172 6.4685739
4.33310811
0.500843491 8.6516211
P-value
5.237E-07
2.126E-09
Lower 95% Upper 95%
52.29868609
100.7716
3.30717557 5.3590406
The correlation coefficient for these data is
A.
B.
C.
D.
E.
20.
1
28
29
0.73
–0.73
0.85
–0.85
unable to be determined from the information provided.
The company would like to determine whether there is a statistically
significant linear correlation between box office gross and number of videos
sold. The appropriate set of hypotheses would be
A.
B.
C.
D.
E.
H0 :   0
HA :   0
H 0 : 1  0
H A : 1  0
H0 : r  0
HA : r  0
H0 :   0
HA :   0
H 0 : 1  0
H A : 1  0
QBM117 Examination - Spring 1999
Page 14 of 14