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Economics 395 Spring 2014 Due May 29th, 2014, before 3 pm Assignment #1 1 Due Thursday, May 29th, by 3 pm in the Economics Office, SS 454. The office is closed during lunch hour, noon – 1 pm. Please remember to do a cover sheet with your name(s), the course number, number all the pages, clearly label which question and part you are answering, and staple your assignment. 1. Express each of the following sums in summation notation and then compute where possible. Let X take the values ????1 = ?2, ????2 = ?1, ????3 = 0, ????4 = 1, ????5 = 2 and Y take the values ????1 = ?1, ????2 = ?0.5, ????3 = 0, ????4 = 1, ????5 = 1.5. a) ????1 + ????2 + ????3 + ????4 + ????5 b) ????3 + ????4 + ????5 c) (????1 ? ????1) + (????2 ? ????2) + (????3 ? ????3) + (????4 ? ????4) + (????5 ? ????5) d) 2(????1 ? ????2) + 2(????2 ? ????3) + 2(????3 ? ????4) e) (????2/????1) + (????3/????2) + (????4/????3) f) (????1 2 ? 2????3) + (????2 2 ? 2????4) + (????3 2 ? 2????5) 2. Show that ? (???????? ? ????? ???? ????=1 ) (???????? ? ?????) = ? ???????? ???? ????=1 ???????? ? ??????????????? 3. An economist was trying to address the growing concern for educators about the number of students who have part-time jobs while they attend high school. To help acquire information, she recorded the average number of hours worked per week and the GPA achieved for 200 students with part time jobs. Below are four equations that were estimated by the economist with the data, where x is the average number of hours worked per week by the student and y is the GPA the student achieved that term. ???? = 3.09 ? 0.15???? ln(????) = 1.56 ? 0.46 ln(????) ???? = 3.83 ? 0.95ln(????) ln(????) = 1.22 ? 0.08???? a) Plot each of the functions for x = 2 to x = 12 on the same graph. Comment on what you observe when you look at the functions on the graph. b) Calculate the slope of each function at x = 7. State the interpretation of the slope. c) Calculate the elasticity of each function at x = 7 and give the interpretation. 4. Let X be a discrete random variable that is the value shown on a single roll of a fair die. a) What is the expected value of X? What is the expected value of X 2 ? b) Find the variance of X. c) What is the probability of X=2 or X=3? d) Find the expected value and variance of W if W = g(X) = 4X + 2.Economics 395 Spring 2014 Due May 29th, 2014, before 3 pm Assignment #1 2 e) Now, assume the die has been weighted so that P(X=1) = 1/3, P(X=2) = P(X=3) = P(X=4) = 1/12, P(X=5) = 1/6, P(X=6) = 1/4. Find the answers to part a) and b) under the new assumption. 5. After watching a number of children playing games in a video arcade, a statistics practitioner estimated the following probability distribution of X, the number of games per visit. x 1 2 3 4 5 6 7 f(x) .05 .05 .15 .25 .30 .15 .05 a) What is the probability that a child will play at least four games? b) What is the mean and variance of the number of games played? c) Suppose that each game costs $2.25. Use the laws of expected value and variance to determine the expected value and variance of the amount of money the arcade takes in per child. d) Determine the probability distribution of the amount of money the arcade takes in per child. e) Use the probability distribution to calculate the mean and variance of the amount of money the arcade takes in per child. f) Are your answers in part c & part e identical? 6. The decision about where to build a new plant is a major one for most companies. One of the factors often considered is the education level of the location’s residents. Census information may be useful in this regard. After analyzing a recent census, a company produced the following joint probabilities: Education Northwest Midwest South West Not a high school graduate 0.0301 0.0318 0.0683 0.0359 High school graduate 0.0711 0.0843 0.1174 0.0608 Some college, no degree 0.0262 0.0410 0.0605 0.0456 Associate’s degree 0.0143 0.0180 0.0248 0.0181 Bachelor’s degree 0.0350 0.0368 0.0559 0.0418 Advanced degree 0.0190 0.0184 0.0269 0.0180 a) What is the probability of a randomly selected individual having an associate’s degree and living in the Northwest? b) Determine the probability that a person living in the West has a bachelor’s degree. c) Find the probability that a high school graduate lives in the South. d) What is the probability that a randomly selected individual lives in the South? e) Are education and location independent?Economics 395 Spring 2014 Due May 29th, 2014, before 3 pm Assignment #1 3 7. A foreman for an injection-modeling firm admits that on 10% of his shifts, he forgets to shut off the injection machine on his line. This cases the machine to overheat, increasing the probability from 2% to 20% that a defective modeling will be produced during the early morning run. a) What proportion of moldings from the early morning run is defective? b) The plant manager randomly selects a molding from the early morning run and discovers the molding is defective. What is the probability that the foreman forgot to shut off the machine the previous night? HINT: Bayes’ Law Formula P(A )P(B| A ) P(A )P(B| A ) . . . P(A )P( B| A ) P(A )P(B| A ) P(A | B) 1 1 2 2 k k i i i ? ? ? ? 8. An investor holds a portfolio consisting of two stocks. He puts 75% of his money in Stock C and 25% into stock F. Stock C has an expected return of RC = 10% and a standard deviation of ?C = 8%. Stock F has an expected return of RF = 15% and a standard deviation of ?F = 15%. The portfolio return is P = 0.75 RC + 0.25 RF. a) What is the expected return on the portfolio? b) Compute the standard deviation of the portfolio if the two stocks’ returns are perfectly positively correlated. c) Compute the standard deviation of the portfolio if the two stocks’ returns have a correlation of 0.5. d) Compute the standard deviation of the portfolio if the two stocks’ returns are uncorrelated. 9. Let Y1, Y2, Y3, and Y4 be independent, identically distributed random variables from a population with a mean ? and a variance ?2. Consider a different estimator of ?: W = 1 12 Y1 + 1 3 Y2 + 5 12 Y3 + 1 6 Y4. This is an example of a weighted average of the Yi. a) Show that W is a linear estimator. b) Is W an unbiased estimator of ?? Show that it is – or it isn’t (E(W) = ?). c) Find the variance of W and compare it to the variance of the sample mean ?????. d) Is W as good an estimator as ?????? Explain your answer. 10. A sample of 23 observations is taken and ????? = 43 and ????? = ???? = 25. a) Test the hypothesis that the mean is greater than 35 at a 1% significance level. Be sure to state the null and alternative hypotheses, give the test statistic and its distribution, indicate the rejection region, including a sketch, state your conclusion, and calculate the p-value for the test. b) Repeat the test in part a) with ????? = ???? = 15. 4 c) Create the 95% confidence interval for the mean when ????? = ???? = 15. d) What is the p-value of your test statistic in part a) and part b)? e) What happens to your test statistic and the p-value when the variance gets smaller?

Economics 395 Spring 2014
Due May 29th, 2014, before 3 pm Assignment #1
1
Due Thursday, May 29th, by 3 pm in the Economics Office, SS 454. The office is
closed during lunch hour, noon – 1 pm.
Please remember to do a cover sheet with your name(s), the course number, number all the
pages, clearly label which question and part you are answering, and staple your
assignment.
1. Express each of the following sums in summation notation and then compute where
possible. Let X take the values ????1 = ?2, ????2 = ?1, ????3 = 0, ????4 = 1, ????5 = 2 and Y take the
values ????1 = ?1, ????2 = ?0.5, ????3 = 0, ????4 = 1, ????5 = 1.5.
a) ????1 + ????2 + ????3 + ????4 + ????5
b) ????3 + ????4 + ????5
c) (????1 ? ????1) + (????2 ? ????2) + (????3 ? ????3) + (????4 ? ????4) + (????5 ? ????5)
d) 2(????1 ? ????2) + 2(????2 ? ????3) + 2(????3 ? ????4)
e) (????2/????1) + (????3/????2) + (????4/????3)
f) (????1
2
? 2????3) + (????2
2
? 2????4) + (????3
2
? 2????5)
2. Show that ? (???????? ? ?????
????
????=1
) (???????? ? ?????) = ? ????????
????
????=1 ???????? ? ???????????????
3. An economist was trying to address the growing concern for educators about the number
of students who have part-time jobs while they attend high school. To help acquire
information, she recorded the average number of hours worked per week and the GPA
achieved for 200 students with part time jobs. Below are four equations that were
estimated by the economist with the data, where x is the average number of hours worked
per week by the student and y is the GPA the student achieved that term.
???? = 3.09 ? 0.15????
ln(????) = 1.56 ? 0.46 ln(????)
???? = 3.83 ? 0.95ln(????)
ln(????) = 1.22 ? 0.08????
a) Plot each of the functions for x = 2 to x = 12 on the same graph. Comment on
what you observe when you look at the functions on the graph.
b) Calculate the slope of each function at x = 7. State the interpretation of the slope.
c) Calculate the elasticity of each function at x = 7 and give the interpretation.
4. Let X be a discrete random variable that is the value shown on a single roll of a fair die.
a) What is the expected value of X? What is the expected value of X
2
?
b) Find the variance of X.
c) What is the probability of X=2 or X=3?
d) Find the expected value and variance of W if W = g(X) = 4X + 2.Economics 395 Spring 2014
Due May 29th, 2014, before 3 pm Assignment #1
2
e) Now, assume the die has been weighted so that P(X=1) = 1/3, P(X=2) = P(X=3) =
P(X=4) = 1/12, P(X=5) = 1/6, P(X=6) = 1/4. Find the answers to part a) and b)
under the new assumption.
5. After watching a number of children playing games in a video arcade, a statistics
practitioner estimated the following probability distribution of X, the number of games per
visit.
x 1 2 3 4 5 6 7
f(x) .05 .05 .15 .25 .30 .15 .05
a) What is the probability that a child will play at least four games?
b) What is the mean and variance of the number of games played?
c) Suppose that each game costs $2.25. Use the laws of expected value and variance to
determine the expected value and variance of the amount of money the arcade takes
in per child.
d) Determine the probability distribution of the amount of money the arcade takes in per
child.
e) Use the probability distribution to calculate the mean and variance of the amount of
money the arcade takes in per child.
f) Are your answers in part c & part e identical?
6. The decision about where to build a new plant is a major one for most companies. One of
the factors often considered is the education level of the location’s residents. Census
information may be useful in this regard. After analyzing a recent census, a company
produced the following joint probabilities:
Education Northwest Midwest South West
Not a high school graduate 0.0301 0.0318 0.0683 0.0359
High school graduate 0.0711 0.0843 0.1174 0.0608
Some college, no degree 0.0262 0.0410 0.0605 0.0456
Associate’s degree 0.0143 0.0180 0.0248 0.0181
Bachelor’s degree 0.0350 0.0368 0.0559 0.0418
Advanced degree 0.0190 0.0184 0.0269 0.0180
a) What is the probability of a randomly selected individual having an associate’s degree
and living in the Northwest?
b) Determine the probability that a person living in the West has a bachelor’s degree.
c) Find the probability that a high school graduate lives in the South.
d) What is the probability that a randomly selected individual lives in the South?
e) Are education and location independent?Economics 395 Spring 2014
Due May 29th, 2014, before 3 pm Assignment #1
3
7. A foreman for an injection-modeling firm admits that on 10% of his shifts, he forgets to
shut off the injection machine on his line. This cases the machine to overheat, increasing
the probability from 2% to 20% that a defective modeling will be produced during the
early morning run.
a) What proportion of moldings from the early morning run is defective?
b) The plant manager randomly selects a molding from the early morning run and
discovers the molding is defective. What is the probability that the foreman forgot
to shut off the machine the previous night?
HINT: Bayes’ Law Formula
P(A )P(B| A ) P(A )P(B| A ) . . . P(A )P( B| A )
P(A )P(B| A )
P(A | B)
1 1 2 2 k k
i i
i
? ? ?
?
8. An investor holds a portfolio consisting of two stocks. He puts 75% of his money in Stock
C and 25% into stock F. Stock C has an expected return of RC = 10% and a standard
deviation of ?C = 8%. Stock F has an expected return of RF = 15% and a standard
deviation of ?F = 15%. The portfolio return is P = 0.75 RC + 0.25 RF.
a) What is the expected return on the portfolio?
b) Compute the standard deviation of the portfolio if the two stocks’ returns are
perfectly positively correlated.
c) Compute the standard deviation of the portfolio if the two stocks’ returns have a
correlation of 0.5.
d) Compute the standard deviation of the portfolio if the two stocks’ returns are
uncorrelated.
9. Let Y1, Y2, Y3, and Y4 be independent, identically distributed random variables from a
population with a mean ? and a variance ?2.
Consider a different estimator of ?:
W =
1
12
Y1 + 1
3
Y2 + 5
12
Y3 + 1
6
Y4.
This is an example of a weighted average of the Yi.
a) Show that W is a linear estimator.
b) Is W an unbiased estimator of ?? Show that it is – or it isn’t (E(W) = ?).
c) Find the variance of W and compare it to the variance of the sample mean ?????.
d) Is W as good an estimator as ?????? Explain your answer.
10. A sample of 23 observations is taken and ????? = 43 and ????? = ???? = 25.
a) Test the hypothesis that the mean is greater than 35 at a 1% significance level. Be
sure to state the null and alternative hypotheses, give the test statistic and its
distribution, indicate the rejection region, including a sketch, state your
conclusion, and calculate the p-value for the test.
b) Repeat the test in part a) with ????? = ???? = 15.
4
c) Create the 95% confidence interval for the mean when ????? = ???? = 15.
d) What is the p-value of your test statistic in part a) and part b)?
e) What happens to your test statistic and the p-value when the variance gets
smaller?

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