Solutions to Statistical Problems on Book Sales, Oil Drilling, Vitamin D, and Investments

Book Store Sales

A book publisher has just published a new book, and sold it in three formats: hardcover, paperback, and electronic. For each format, for the first three months, the book was sold at full price, and after that at a discount. The data from 1000 customers is attached in the data file, “Book Data.xlsx”. The following table shows the price of each transaction type. If your answer does not match exactly, choose the one that is closest to your answer.

Q1. What fraction of books were sold in paperback?

1. 14 (14%)
2. 20 (20%)
3. 34 (34%)
4. 48 (48%)

Q2. If a customer who purchased the book is randomly selected, what is the probability that customer paid more than $14 for the book?

1. 10 (10%)
2. 22 (22%)
3. 28 (28%)
4. 30 (30%)

Q3. What is the probability that a customer paid full price for the book given that they purchased the book in hardcover?

1. 02 (2%)
2. 08 (8%)
3. 15 (15%)
4. 80 (80%)

Q4. What is the probability that the book was sold for more than $15 given that it was sold at full price?

1. 28 (28%)
2. 54 (54%)
3. 58 (58%)
4. 93 (93%)

Q5. What is the average (mean) price that the book was sold?

11. $11.15
12. $12.44
13. $15.33
14. $17.50

Q6. What is the probability that the book was sold in the electronic format or it was sold for less than $15?

1. 25 (25%)
2. 44 (44%)
3. 56 (56%)
4. 70 (70%)

Q7. Which of the following pairs of events are mutually exclusive?

1. (Book sold at discount) and (Book sold in paperback)
2. (Book sold in paperback) and (Book sold for more than $17)
3. (Book sold in electronic format) and (Book sold for more than $9)
4. None of the above.

Numbers may vary slightly due to rounding choices. Choose the answer that is closest to your own.

You work for a large multinational oil company, DrillAll, and your company has just acquired leases for 25 sites dispersed at various places in a region of South-Central Pacific. Based on initial exploration and analysis, you believe that each site has a 11% probability of being “good" (i.e. containing crude oil that can be commercially viable). Due to the dispersion of the sites, they can be assumed to be independent of each other.

Q1. Let X be a random variable that denotes the number of good sites among the 25 sites. What are the mean and standard deviation of X?

2. E[X] = 2.75, Stdev[X]= 2.448
3. E[X] = 2.75, Stdev[X]= 1.564
4. E[X] = 22.75, Stdev[X]= 1.564
5. E[X] = 11, Stdev[X]= 5

Q2. What is the probability that 5 or more sites are good?

1. 950
2. 984
3. 133
4. 050

Q3. What is the probability that there is no good site among the 25 sites?  In other words, what is the probability that the mission is a complete bust?

1. Approximately 0.000
2. 054
3. 222
4. 400

Q4. Drilling at each site costs $4M. Suppose that the revenue generated from a good site is about $50M while no revenue will be generated from a bad site. Then, what is the expected profit from this mission of drilling all 25 sites? For simplicity, assume no other cost is involved.

137. $137.5M
138. $126.5M
139. $ 100 M
140. $37.5M

Drill, Drill, Drill

Q5. Drilling each site costs $4M. Suppose that the revenue generated from a good site is about $50M while no revenue will be generated from a bad site. Then, what is the standard deviation of the profit from this mission of drilling all 25 sites? For simplicity, assume no other cost is involved.

21. $-21.8M
22. $78.2M
23. $122.4 M
24. $6118.8M

Numbers may vary slightly due to rounding choices. Choose the answer that is closest to your own.

Vitamin D deficiency among children in the US almost disappeared in the ’70s. However, in recent years, a study conducted by a pharmaceutical company claims that 30% of children have vitamin D deficiency due to lack of outdoor activities. A consumer advocacy group wants to conduct an independent study to verify the claim.

The group selected a random sample of 200 children in the United States.

Q1. If the claim made by the study is actually true (the proportion of children with vitamin D deficiency is indeed 30%, p=0.3), what distribution should the sample proportion computed from a sample of 200 children should follow? 

a. The sample proportion is always equal to p=0.3. There is no need to discuss the distributions of the sample proportion and the resultant sampling error.

b. The sample proportion, , varies depending on who belongs to the sample. Since n=200 is large enough, the sample proportion approximately follows a normal distribution with mean p=0.3, and standard error (which is the standard deviation of the sample proportion) 0.0324.      

c. The sample proportion, , varies depending on who belongs to the sample.  But, the distribution of the sample proportion approximately follows a normal distribution with mean p=0.3 and standard deviation 0.458.

d. The sample proportion, , varies depending on who belongs to the sample.  The distribution of the sample proportion approximately follows a normal distribution with mean p=0.3 and standard error 0.0011.

Q2. If the claim made by the study is actually true (p=0.3, the proportion of children with vitamin D deficiency is indeed 30%),  what is the probability that the sample proportion, ,  will exceed 0.35 (35%)?

1. 050
2. 061
3. 100
4. 939

Q3. If the claim made by the study is actually true the proportion of children with vitamin D deficiency is indeed 30%, p=0.3), what is the probability that the sampling error (= sample proportion – true proportion) is greater than 0.05?

1. 050
2. 061
3. 100
4. 939
5. The information about the distribution of the sampling error is insufficient. Thus we cannot compute the probability.

A private equity firm, Jupiter, is evaluating to choose one of two alternative investments to invest in long-term. Although exact returns are random, each investment’s return is well diversified enough that its distribution is estimated to follow a normal distribution. Jupiter does not have enough capital, thus it can choose only one investment option. Each investment costs about $10M of capital. We assume that the performances of the two funds are independent of each other.

The first investment (The Nickel fund) has a expected return of $2M with a standard deviation of $0.3M ($300K).

The second investment (The Dime fund) has a expected return of $2.5M with a standard deviation of $0.7M ($700K).

Q1. If the decision is based solely on the expected return, which investment option should Jupiter choose? If the decision is based solely on minimizing the standard deviation, which investment option should Jupiter choose?  

Based on expected return:    _________________________

Based on standard deviation: _________________________

Q2.  In addition to the standard deviation, one of the criteria Jupiter is considering is the likelihood the return from each investment is less than $1.5M. If Jupiter wants to choose the investment option that minimizes the chance that the return is less than $1.5M, which investment option should Jupiter choose? (For full credit, provide a brief justification.)