COMPREHENSIVE EXAMINATION
Managerial Statistics
Part A. TRUE OR FALSE.
1. The date when a production line in a factor is out-of-control will be measured with a ratio scale.
ANSWER: False
2.
The amount of time a student spent studying for an exam is an example of a continuous
variable.
ANSWER: True
3.
As a general rule, an observation is considered an extreme value if its Z score is less than -3.
ANSWER: True
4.
The Z score of an observation measures how many standard deviations is the value from the
mean.
ANSWER: True
5.
An economics professor bases his final grade on homework, two midterm examinations, and a
final examination. The homework counts 10% toward the final grade, while each midterm
examination counts 25%. The remaining portion consists of the final examination. If a student
scored 95% in homework, 72% on the first midterm examination, 96% on the second midterm
examination, and 72% on the final, his final average is 79.8%.
ANSWER: False
6.
The median of the values 3.4, 4.7, 1.9, 7.6, and 6.5 is 1.9.
ANSWER: False
7.
The number of males selected in a sample of 5 students taken without replacement from a
class of 9 females and 18 males has a binomial distribution.
ANSWER: False
8.
The question: “How much did you make last year rounded to the nearest hundreds of dollars?”
will most likely result in measurement error.
ANSWER: True
9.
The analysis of variance (ANOVA) tests hypotheses about the population variance.
ANSWER: False
10. If P(A and B) = 0, then A and B must be mutually exclusive.
ANSWER: True
Part B. MULTIPLE CHOICE
11. The closing price of a company’s stock tomorrow can be lower, higher or the same as today’s
closed. Without any prior information that may affect the price of the stock tomorrow, the
probability that it will close higher than today’s close is 1/3. This is an example of using
which of the following probability approach?
a. A priori classical probability
b. Empirical classical probability
c. Subjective probability
d. Conditional probability
ANSWER: a
12. Scientists in the Amazon are trying to find a cure for a deadly disease that is attacking the
rain forests there. One of the variables that the scientists have been measuring involves the
diameter of the trunk of the trees that have been affected by the disease. Scientists have
calculated that the average diameter of the diseased trees is 42 centimeters. They also know
that approximately 95% of the diameters fall between 32 and 52 centimeters and almost all
of the diseased trees have diameters between 27 and 57 centimeters. When modeling the
diameters of diseased trees, which distribution should the scientists use?
a. Uniform distribution
b. Binomial distribution
c. Normal distribution
d. Exponential distribution
ANSWER: c
13. Which of the following sampling methods will more likely be susceptible to ethical violation?
a. Simple random sample
b. Cluster sample
c. Convenience sample
d. Stratified sample
ANSWER: c
14. The Dean of Students mailed a survey to a total of 400 students. The sample included 100
students randomly selected from each of the freshman, sophomore, junior, and senior
classes on campus last term. What sampling method was used?
a. Simple random sample
b. Systematic sample
c. Stratified sample
d. Cluster sample
ANSWER: c
15. Which of the following types of samples can you use if you want to make valid statistical
inferences from a sample to a population?
a. A judgment sample
b. A quota sample
c. A chunk
d. A probability sample
ANSWER: d
16. In a one-way ANOVA, if the computed F statistic exceeds the critical F value we may
a. reject H0 since there is evidence all the means differ.
b. reject H0 since there is evidence of a treatment effect.
c. not reject H0 since there is no evidence of a difference.
d. not reject H0 because a mistake has been made.
ANSWER: b
17. Interaction in an experimental design can be tested in
a. a completely randomized model.
b. a randomized block model.
c. a two-factor model.
d. all ANOVA models.
ANSWER: c
18. If we are testing for the difference between the means of 2 related populations with samples
of n1 = 20 and n2 = 20, the number of degrees of freedom is equal to
a. 39.
b. 38.
c. 19.
d. 18.
ANSWER: c
DRUNK DRIVING
Mothers Against Drunk Driving is a very visible group whose main focus is to educate the public
about the harm caused by drunk drivers. A study was recently done that emphasized the
problem we all face with drinking and driving. Four hundred accidents that occurred on a
Saturday night were analyzed. Two items noted were the number of vehicles involved and
whether alcohol played a role in the accident. The numbers are shown below:
Did alcohol play a
role?
Yes
No
Totals
Number of Vehicles
Involved
1
2
3
50
25
75
100
175
275
20
30
50
19. What proportion of accidents involved alcohol or a single vehicle?
a. 25/400 or 6.25%
b. 50/400 or 12.5%
c. 195/400 or 48.75%
Totals
170
230
400
d. 245/400 or 61.25%
ANSWER: c
20. Referring to Table 4-1, given that multiple vehicles were involved, what proportion of
accidents involved alcohol?
a. 120/170 or 70.59%
b. 120/230 or 52.17%
c. 120/325 or 36.92%
d. 120/400 or 30%
ANSWER: c
Part C. FILL IN THE BLANKS
The table below contains the opinions of a sample of 200 people broken down by gender about the
latest congressional plan to eliminate anti-trust exemptions for professional baseball.
Female
Male
Totals
For
38
12
50
Neutral
54
36
90
Against
12
48
60
Totals
104
96
200
21. Out of the males in the sample, ________ percent were for the plan.
ANSWER: 12.50%
22. ________ percent of the 200 were females who were against the plan.
ANSWER: 6%
23. _______ percent of the 200 were males who were not against the plan.
ANSWER: 24%
24. if the sample is a good representation of the population, we can expect _______ percent of the
population will be for the plan.
ANSWER: 25%
25. Suppose that patrons of a restaurant were asked whether they preferred beer or whether they
preferred wine. 70% said that they preferred beer. 60% of the patrons were male. 80% of the
males preferred beer. The probability a randomly selected patron prefers wine is __________.
ANSWER: 0.30
26. The amount of time necessary for assembly line workers to complete a product is a normal
random variable with a mean of 15 minutes and a standard deviation of 2 minutes. So, 70% of
the products would be assembled within __________ minutes.
ANSWER: 16.0488 using Excel or 16.04 using Table E.2
27. Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1. So
96% of the possible Z values are between __________ and __________ (symmetrically
distributed about the mean).
ANSWER: -2.05 and 2.05 or -2.06 and 2.06
28. The true length of boards cut at a mill with a listed length of 10 feet is normally distributed with
a mean of 123 inches and a standard deviation of 1 inch. What proportion of the boards will be
less than 124 inches?
ANSWER: 0.8413
29. If you were constructing a 99% confidence interval of the population mean based on a sample
of n=25 where the standard deviation of the sample s = 0.05, the critical value of t will
be______
ANSWER: 2.7969
30. In the construction of confidence intervals, if all other quantities are unchanged, an increase in
the sample size will lead to a
interval.
ANSWER: narrower
HOME IMPROVEMENT STORE
A major home improvement store conducted its biggest brand recognition campaign in the
company’s history. A series of new social media advertisements featuring well-known entertainers
and sports figures were launched. A key metric for the success of advertisements is the proportion
of viewers who “like the ads a lot”. A study of 1,189 adults who viewed the ads reported that 230
indicated that they “like the ads a lot.” The percentage of a typical advertisement receiving the “like
the ads a lot” score is believed to be 22%. Company officials wanted to know if there is evidence
that the series of advertisements are less successful than the typical ad (i.e. if there is evidence that
the population proportion of “like the ads a lot” for the company’s ads is less than 0.22) at a 0.01
level of significance.
31. The parameter the company officials is interested in is ___________________.
ANSWER: the proportion of viewers who “like the ads a lot”
32. State the null hypothesis for this study.
ANSWER: H 0 : p  0.22
33. State the alternative hypothesis for this study.
ANSWER: H1 : p  0.22
34. What critical value should the company officials use to determine the rejection region?
ANSWER: 2.3263
35. The null hypothesis will be rejected if the test statistics is _______________.
ANSWER: less than 2.3263
36. State the decision for this problem.
ANSWER: Do not reject the null hypothesis.
37. State the conclusion for this problem.
ANSWER: The company officials can conclude that there is sufficient evidence to show that the series of
television advertisements are less successful than the typical ad using a level of significance of 0.05.
Part D. ANALYSIS and INTERPRETATION
Aria’s Bakeshop supplies pastry products to a number of supermarkets in the city and wishes to
study the effect of the shelf display height on monthly sales measured in number of cases. Shelf
display height has three levels – Bottom, Middle and Top, set as treatments. For each shelf height, 6
supermarket, the experimental units, displays the products using its assigned shelf height for a
month. At the end of the month, sales, the response variable, at the 18 participating stores are
recorded as:
Bottom
58.2
53.7
55.8
55.7
52.5
58.9
Middle
73.0
78.1
75.4
76.2
78.4
82.1
Top
52.4
49.7
50.9
54.0
52.1
49.9
Assume that the amount of sales for each display height is randomly selected from the population of
all sales amounts that could be obtained at supermarkets at that display height. Shown is the Excel
result of the test conducted:
Anova: Single Factor
SUMMARY
Groups
Bottom
Middle
Top
Count
6
6
6
ANOVA
Source of Variation
Between Groups
Within Groups
SS
2273.88
92.4
Total
2366.28
38. What is the null hypothesis?
Sum
Average Variance
334.8
55.8
6.136
463.2
77.2
9.628
309
51.5
2.716
df
2
15
17
MS
F
P-value
1136.94 184.5682 2.74E-11
6.16
F crit
3.68232
ANSWER: Ho: µ1 = µ2
39. What is the total variation equal to?
ANSWER: 2366.28
40. What is the decision about the null hypothesis?
ANSWER: Reject the null hypothesis
41. Interpret the indication of the computed p-value.
ANSWER: The test failed to accept the null hypothesis since there is a significant difference between
means of bottom, middle and top shelves at alpha = 0.05.
42. State the conclusion of the study conducted by Aria’s Bakeshop.
ANSWER: The sales are significantly different for different shelf height levels. Therefore, the
shelf height levels affect the sales of Aria’s Bakeshop products.
Refer to MRA Result A shown below to answer the following questions:
43. What is the part of the variation in Selling Price that can be explained by the regression
equation? ANSWER: 6635.481
44. What is the part of the variation in Selling Price that is not explained by the regression
equation? ANSWER: 1018.239
45. What is the total variation equal to? 7653.730
46. What is the proportion of the variability in Selling Price that can be explained by the regression
equation? ANSWER: 86.7% of the variation in selling price can be explained by the variation
due to the number of rooms and the neighborhood.
47. What is the proportion of the variability in Selling Price that are due to other factors? ANSWER:
13.3% of the variation in Selling price is due to factors other than number of rooms and
neighborhood.
48. What is the variation around the prediction line? ANSWER: 7.739
49. What is the regression equation? ANSWER: Selling Price = 243.7371 + 9.2189 * (the number of
rooms) + 12.6367 * (the type of neighborhood)
50. Predict the price of a house with 9 rooms and located in the west side neighborhood. Make
sure to include the units in your answer. ANSWER: Selling Price = 243.7371 + (9.2189 * 9) +
(12.6367 * 1) = 339.3439 * 1000 = $ 339,343.90