John is looking to invest $10000, and has a choice of purchasing two different financial assets, A and B. The annual return on A is known to follow a normal distribution with a mean of 8% and a standard deviation of 10%. The annual return on B is also known to follow a normal distribution with a mean of 6% and a standard deviation of 5%. The return on A is known to be independent of the return on B.
a) What is the probability that A will have a return higher than 10%?
A) 0.1587
B) 0.3413
C) 0.4207
D) 0.8413

b) Find the probability that each asset will make a loss.
A) A: 0.5, B: 0.3085
B) A: 0.5, B: 0.1587
C) A: 0.8413, B: 0.5
D) A: 0.3085, B: 0.1587

c) Suppose that in a given year, we know that A had a return of 5%. What is the probability that the return on B is lower than A in that year, but higher than 0%?
A) 0.1587
B) 0.4207
C) 0.3413
D) 0.3085

d) Suppose John decides to invest 40% of his money in A and 60% in B. What is the expected return on his investment? What is the probability that the value of his portfolio will exceed $11000 at the end of the year?
A) Expected return: 6.4%, Probability: 0.8413
B) Expected return: 6.4%, Probability: 0.5
C) Expected return: 6.8%, Probability: 0.1587
D) Expected return: 6.8%, Probability: 0.8413

e) John tells you that while he would like a high expected return, the most important thing to him is to minimize the probability of making a loss. How would you advise he invests his money? Give reasons for your advice.
A) Invest 100% in A as it has higher returns.
B) Invest 100% in B as it has lower standard deviation.
C) Invest 60% in A and 40% in B to balance risk and return.
D) Invest 40% in A and 60% in B to balance risk and return.

It is known that amounts of money spent on textbooks in a year by students on a particular campus follow a normal distribution with mean $280 and standard deviation $60.
a) What is the probability that a randomly chosen student will spend less than $300 on textbooks in a year?
A) 0.6915
B) 0.3085
C) 0.8413
D) 0.1587

b) What is the probability that a randomly chosen student will spend between $200 and $300 on textbooks in a year?
A) 0.4207
B) 0.8413
C) 0.1587
D) 0.3085

c) Suppose a sample of 25 students is collected. How will the sample mean be distributed? What is its mean and standard deviation?
A) Normal distribution, mean: $280, standard deviation: $60/√25
B) Normal distribution, mean: $280, standard deviation: $60
C) Normal distribution, mean: $280, standard deviation: $15
D) Normal distribution, mean: $25, standard deviation: $280

d) What is the probability that the sample mean will be between $275 and $285?
A) 0.4207
B) 0.3413
C) 0.1587
D) 0.6915