Question 1: Random walks and stock prices. The “random walk” theory of securitie
ID: 3225922 • Letter: Q
Question
Question 1:
Random walks and stock prices. The “random walk” theory of securities prices holds that price movements in disjoint time periods are independent of each other. Suppose that we record only whether the price is up or down each year, and that the probability that our portfolio rises in price in any one year is 0.65. (This probability is approximately correct for a portfolio containing equal dollar amounts of all common stocks listed on the New York Stock Exchange.)
(a) What is the probability that our portfolio goes up for 3 consecutive years?
(b) If you know that the portfolio has risen in price 2 years in a row, what probability do you assign to the event that it will go down next year?
(c) What is the probability that the portfolio’s value moves in the same direction in both of the next 2 years?
Question 2:
An ancient Korean drinking game. An ancient Korean drinking game involves a 14-sided die. The players roll the die in turn and must submit to whatever humiliation is written on the up-face: something like “Keep still when tickled on face.” Six of the 14 faces are squares. Let’s call them A, B, C, D, E, and F for short. The other eight faces are triangles, which we will call 1, 2, 3, 4, 5, 6, 7, and 8. Each of the squares is equally likely. Each of the triangles is also equally likely, but the triangle probability differs from the square probability. The probability of getting a square is 0.72. Give the probability model for the 14 possible outcomes.
Question 3:
ACT scores of high school seniors. The scores of high school seniors on the ACT college entrance examination in 2013 had mean = 20.8 and standard deviation = 4.8. The distribution of scores is only roughly Normal.
(a) What is the approximate probability that a single student randomly chosen from all those taking the test scores 23 or higher?
(b) Now take an SRS of 25 students who took the test. What are the mean and standard deviation of the sample mean score of these 25 students?
(c) What is the approximate probability that the mean score of these students is 23 or higher?
(d) Which of your two Normal probability calculations in (a) and (c) is more accurate? Why?
Question 4:
The effect of sample size on the standard deviation. Assume that the standard deviation in a very large population is 100.
(a) Calculate the standard deviation for the sample mean for samples of size 1, 4, 25, 100, 250, 500, 1000, and 5000.
(b) Graph your results with the sample size on the x axis and the standard deviation on the y axis.
(c) Summarize the relationship between the sample size and the standard deviation that you showed in your graph.
Question 5:
A quality control technician is checking the weights of a product. She takes a random sample of 8 units and weighs each unit. The observed weights are shown below. Assume the population has a normal distribution.
Weight
50
48
55
52
53
46
54
50
Provide a 95% confidence interval for the mean weight of the units.
Question 6:
The average monthly electric bill of a random sample of 256 residents of a city is $90 with a standard deviation of $24.
a.
Construct a 90% confidence interval for the mean monthly electric bills of all residents.
b.
Construct a 95% confidence interval for the mean monthly electric bills of all residents.
Question 7:
A researcher is interested in determining the average number of years employees of a company stay with the company. If past information shows a standard deviation of 7 months, what size sample should be taken so that at 95% confidence the margin of error will be 2 months or less?
Weight
50
48
55
52
53
46
54
50
Explanation / Answer
Question1:
(a) What is the probability that our portfolio goes up for 3 consecutive years?
Answer :
P(price up AND price up AND price up) = 0.653 =(0.65)(0.65)(0.65)= 0.274625
(b) If you know that the portfolio has risen in price 2 years in a row, what probability do you assign to the event that it will go down next year?
Answer:
Let A = portfolio has risen 2 years in a row
B = portfolio will go down next year.
P(B/A) = P(A & B) / P(A)
P(A) = (0.65)(0.65)
P( A & B) = (0.65)(0.65)(0.35) =
P(B/A) = (0.65)(0.65)(0.35) / (0.65)(0.65) = 0.35
(c) What is the probability that the portfolio’s value moves in the same direction in both of the next 2 years?
Answer :
P(down And down OR up And up) = P(down And down) + P(up And up)
= 0.352 + 0.652
= 0.545
Hope this will be helpful. Thanks and God Bless You:-)
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