Exercises - Columbia Business School

Fall 2001 B6014: Managerial Statistics

Exercises

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

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Exercises Descriptive Statistics

Fall 2001 B6014: Managerial Statistics

Professor Paul Glasserman 403 Uris Hall

# Occurrences

8 7 6 5 4 3 2 1 0

01234567 Values

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 fifth student did not report a starting salary. Choose one of the following:

(a) The median starting salary for all five students could be anywhere between $95,000 and $118,000.

(b) The median starting salary for all five students is $106,000. (c) The median starting salary for all five students is $106,500. (d) The median starting salary for all five 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 figure 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

0.2

Frequency

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

# observations

20 18 16 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.

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