Economics 395 Assignment #1

Economics 395 Spring 2014 Due May 29th, 2014, before 3 pm Assignment #1

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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 𝑥1 = −2, 𝑥2 = −1, 𝑥3 = 0, 𝑥4 = 1, 𝑥5 = 2 and Y take the values 𝑦1 = −1, 𝑦2 = −0.5, 𝑦3 = 0, 𝑦4 = 1, 𝑦5 = 1.5.

a) 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 + 𝑥5 b) 𝑦3 + 𝑦4 + 𝑦5 c) (𝑥1 − 𝑦1) + (𝑥2 − 𝑦2) + (𝑥3 − 𝑦3) + (𝑥4 − 𝑦4) + (𝑥5 − 𝑦5) d) 2(𝑥1 ∗ 𝑦2) + 2(𝑥2 ∗ 𝑦3) + 2(𝑥3 ∗ 𝑦4) e) (𝑥2/𝑦1) + (𝑥3/𝑦2) + (𝑥4/𝑦3)

f) (𝑥1 2

∗ 2𝑦3) + (𝑥2 2 ∗ 2𝑦4) + (𝑥3

2 ∗ 2𝑦5)

2. Show that ∑ (𝑥𝑖 − �̅� 𝑛 𝑖=1 ) (𝑦𝑖 − �̅�) = ∑ 𝑥𝑖

𝑛 𝑖=1 𝑦𝑖 − 𝑛𝑥𝑦̅̅ ̅

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.

𝑦 = 3.09 − 0.15𝑥 ln(𝑦) = 1.56 − 0.46 ln(𝑥)

𝑦 = 3.83 − 0.95ln (𝑥) ln(𝑦) = 1.22 − 0.08𝑥

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 X2? 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

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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

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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

)A|B(P)A(P...)A|B(P)A(P)A|B(P)A(P

)A|B(P)A(P )B|A(P

kk2211

ii 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 σC = 8%. Stock F has an expected return of RF = 15% and a standard

deviation of σF = 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 μ and a variance σ2. Consider a different estimator of μ:

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 μ? 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 �̂� = 𝑠 = 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 �̂� = 𝑠 = 15.

Economics 395 Spring 2014 Due May 29th, 2014, before 3 pm Assignment #1

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c) Create the 95% confidence interval for the mean when �̂� = 𝑠 = 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?

11. Random samples from two normal populations produced the following statistics:

𝑠1 2 = 45, 𝑛1 = 45, 𝑠2

2 = 25, 𝑛2 = 40

a) Can we infer at the 10% significance level that the two population variances differ?

b) Now, repeat part a) changing the sample sizes, n1 and n2, to 25. What happens to the test statistic and the critical values when the sample size decreases?

12. The Excel file posted with this assignment contains sample data, from 2004, on the hourly wage rate of a sample of individuals, along with what province the person lives in.

You can use Excel or Stata for this question.

a) Calculate the mean hourly wage and variance for workers over all the data, the Canadian average. Now, calculate the mean hourly wage and variance for

unionized workers for each province. Using a table summarize the mean hourly

wage, the variance, and the sample size for each province and for all the data (the

Canadian average).

b) Using the data for New Brunswick, test the hypothesis that the mean hourly wage for workers is greater than $17.15 at the 5% 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.

c) Using the data for New Brunswick, test the hypothesis that the mean hourly wage for workers is equal to $17.15 at the 5% 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.

d) Comment on your results from parts b) & c). e) Assume that the hourly wage rates for workers in Alberta and Quebec are

normally distributed with means μA and μQ, and variances σA 2 and σQ

2

respectively. Assume the hourly wage rates in Alberta and Quebec are

independent of each other. Test the hypothesis that the variances σA 2 and σQ

2 are

equal, against the alternative that the variance in Alberta is larger, at a 5%

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.

f) Test the hypothesis that the mean hourly wage of workers in Alberta and Quebec are equal at a 5% significance level. Use your conclusion in part e) to do the

appropriate version of the test.