Exercises
Exercises
Fall 2001
B6014: Managerial Statistics
Professor Paul Glasserman
403 Uris Hall
1. Descriptive Statistics
2. Probability and Expected Value
3. Covariance and Correlation
4. Normal Distribution
5. Sampling
6. Confidence Intervals
7. Hypothesis Testing
8. Regression Analysis
8
Exercises
Descriptive Statistics
Fall 2001
B6014: Managerial Statistics
Professor Paul Glasserman
403 Uris Hall
8
# Occurrences
7
6
5
4
3
2
1
0
0
1
2
3
4
Values
5
6
7
Figure 1: Histogram for Problem 1
1. Find the median of the data in Figure 1.
2. Find the standard deviation of the data in Figure 1.
3. Five students from the 1999 MBA class took jobs in rocket science after graduation. Four
of these students reported their starting salaries: $95,000, $106,000, $106,000, $118,000.
The ?fth student did not report a starting salary. Choose one of the following:
(a) The median starting salary for all ?ve students could be anywhere between $95,000
and $118,000.
(b) The median starting salary for all ?ve students is $106,000.
(c) The median starting salary for all ?ve students is $106,500.
(d) The median starting salary for all ?ve students could be greater than $118,000.
4. The observations X1 , . . . , Xn have a mean of 52, a median of 52.1, and a standard deviation
of 7. Eight percent of the observation are greater than 66; 7.9% of the observations are
below 38. Based on this information, which of the following statements best describes the
data?
9
2
1.5
1
0.5
0
-2
-1
0
1
2
-0.5
-1
-1.5
-2
Figure 2: Scatter plot for question 5
(i) The distribution has positive skew.
(ii) The distribution has negative skew.
(iii) The distribution has high kurtosis.
(iv) The distribution conforms to a normal distribution.
5. Consider the data in the scatter plot of Figure 2. The correlation between the X and Y
values in the ?gure is closest to
(i) 0.2
(ii) ?0.2
(iii) 1
(iv) ?1
(v) 0
6. The observations X1 , . . . , Xn have a mean of 50 and a standard deviation of 7. Which of
the following statements is guaranteed to be true according to Chebyshev¡¯s rule? (Write
¡°True¡± or ¡°False¡± next to each.)
(i) At least 75% of the observations are between 36 and 64
(ii) At least 80% of the observations are between 34 and 66
(iii) At least 88.9% of the observations are between 31 and 73
(iv) Fewer than 15% of the observations are below 30
7. Suppose the observations X1 , X2 , . . . , Xn have mean 10. Suppose that exactly 75% of the
observations are less than or equal to 15. According to Chebyshev¡¯s rule, what is the
smallest possible value of the population standard deviation of these observations?
10
0.25
Frequency
0.2
0.15
0.1
0.05
0
0
10
20
30
40
50
60
70
80
90
100
Price
Figure 3: Histogram of bond prices at default, 1974-1995. (Source: Moody¡¯s Investor Services.)
8. Which of the following best describes the data in Figure 3? (Base your answer on the
appearance of the histogram. You do not need to do any calculations. Select just one
statement below and complete the one you select.)
(a) The mean is greater than the median because
(b) The median is greater than the mean because
(c) The mean and median are roughly equal because
9. One proposal that has received little attention from Major League Baseball is to pay
pitchers according to the following rule: each pitcher receives a base salary of $4.25 million,
minus $0.25 million times his earned run average (ERA). (A lower ERA is associated with
better performance.) If this rule were adopted, what would be the correlation between a
pitcher¡¯s earnings and ERA? (Assume that the ERA cannot exceed 17, so this rule never
results in negative earnings. You may also assume a standard deviation of 1.2 for ERA.)
10. Using the data in Figure 4, answer both (a) and (b) below, providing a numerical value
for each.
(a) The mean of the data in the histogram is
(b) The median of the data in the histogram is
11. Cluster ¦· had exams in Finance and Marketing last week. All 60 students in the cluster
took both exams. The results were as follows:
? Finance: mean = 25, standard deviation = 2
? Marketing: mean = 75, standard deviation = 12
? Correlation between score in Finance and same student¡¯s score in Marketing = 0.84
Mary, a student in Cluster ¦·, scored a 30 in Finance and a 90 in Marketing. We are
interested in comparing her performance on the two exams relative to the rest of the
class. In particular, we would like to make a statement about which of her scores ranked
higher compared to the other scores on the same exam. Select one of the choices below
and complete the statement you select.
11
20
18
16
# observations
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
9
10
value
Figure 4: Histogram for Problem 10
(i) Mary¡¯s score in Finance probably ranks higher than her score in Marketing because
(ii) Mary¡¯s score in Marketing probably ranks higher than her score in Finance because
(iii) Mary¡¯s scores on the two exams probably rank about equally high because
(iv) We cannot make any comparison between the two scores because
12. Seven students from the 1998 MBA class took jobs in brain surgery after graduation. Five
of the students reported their starting salaries: $55,000, $90,250, $90,250, $95,500, and
$105,000. Choose one of the following:
(a) Based on the information given, the largest possible value of the median starting
salary for all seven students is
(b) Based on the information given, it is not possible to put an upper limit on the median
starting salary for all seven students.
12
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