The temperature in January is estimated to have a mean of 34 and a standard

1. The temperature in January is estimated to have a mean of 34 and a standard

deviation of 6, in degrees Fahrenheit. Sally estimates that her heating bill can

be predicted using the following formula:

X = 300 − 5 × temp,

where temp is temperature. Find how much she is expected to pay in January.

Moreover, obtain the variance and standard deviation of the heating bill.

2. An instructor graded a large number of midterm exams and she considers that

the test scores are normally distributed with a mean of 70 and a standard

deviation of 10.

(a) What is the portion of students obtaining scores between 85 and 95?

(b) What is the score needed to be at the top 10% of the class?

3. Response time at an online site can be modeled with an exponential distribution

with a mean service of 5 minutes. John knows this and it is not sure to wait

for an answer. What is the probability that for the reply to John’s request

(a) Will take longer than 10 minutes?

(b) Shorter than 10 minutes?

(c) If the mean service is now 2 minutes, what is the probability that it takes

longer than 10 minutes?

4. James W. has a portfolio that includes 20 shares of Disney and 30 shares of

Amazon. The price of Disney stock is normally distributed with a mean of 25

and a variance of 80. The price of Amazon is also normally distributed with

a mean of 40 and a variance of 119. James finds out that these stocks are

negatively correlated, with δ = −0.4.

Economics 391 (Spring 2013) Professor Lamarche, University of Kentucky

(a) Find the mean and standard deviation of James W.’s portfolio.

(b) Would you advice James to sell Amazon and buy Disney?

(c) What is the probability that the value of the portfolio is greater than

$2,000?

(d) What is the mean and standard deviation of James’ portfolio if stocks are

not correlated?

5. Use the computer to calculate the following probabilities:

(a) P(t 64 > 2.12)

(b) P(t 27 < 1.90)

(c) P(t 121 < 1)

(d) P(F 7.20 > 2.5)

(e) P(F 34.62 > 1.8)

(f) P(χ 2

3

> 1.61)

6. Suppose you have an investment A whose return is normally distributed with

mean 8% and standard deviation of 5 %. An alternative investment B gives an

the same return with a standard deviation of 8%.

(a) What is the probability of losing money in investments A and B?

(b) Would you rather construct a portfolio that uses A and B (e.g., C =

10A + 10B)? Why?