answer only question 5

Economics 250 Introductory Statistics

Exercise 2 Due Thursday 3 March in class and on paper

Instructions: There is no drop box and this exercise can be submitted only in class. No late submissions are possible. Show intermediate steps (such as formulas) in your work both for ease of review and part marks in case of error. Do the exercise on your own and submit only your own work. Write in pen (or type) and not in pencil. If you are late to class please submit your work at the end of the meeting.

1. Suppose that we know that, in a population, spells of unemployment are normally distributed with a mean of 220 days and a standard deviation of 6 days.

(a) What is the probability that a spell lasts more than 226 days?

(b) Researchers collect a random sample of 4 unemployment spells. What is the probability that the average spell lasts more than 226 days?

2. Suppose that researchers devise an indicator of business cycles with 0 indicating a recession and 1 indicating no recession. In two adjacent years, the indicators are jointly distributed like this:

This Year 0 1

0 0.5 0.1 Next Year 1 0.1 0.3

(a) Find the marginal distribution for the indicator next year. Then find the mean and standard deviation using that distribution. (Sometimes these are called the unconditional mean and standard deviation.)

(b) Suppose that you observe that the indicator takes the value 1 this year. Find the conditional distribution for the indicator next year. Then find the conditional mean and standard deviation.

(c) Are the two random variables independent? Briefly justify your answer.

3. Suppose that 2% of athletes in a competition are cheating by taking banned stimulants. A test is available that yields evidence of cheating (i.e. a positive test) 8% of the time for athletes who are not cheating and 70% of the time for athletes who are cheating.

(a) What is the probability of a randomly selected athlete testing positive?

(b) If an athlete tests positive what is the probability she or he was cheating?

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4. Suppose that a training program succeeds in raising any worker’s wage with a proba- bility of 0.60.

(a) If 8 workers participate in the program then what is the probability that at least half of them (i.e. 4 or more) experience a wage increase?

(b) What is the probability that at most 3 of them experience a wage increase?

(c) If 80 workers participate then what is the probability that at least half of them ex- perience a wage increase? (Hint: Think of this sample size as large and do not use the binomial formula.)

(d) The agency funding the program does not yet know its budget or what the sample size will be. Given the success rate in the population, describe the distribution of the sample success rate (or proportion of successes) for a sample size n (assuming n is large).

5. Statistics are central to managing an investment portfolio. Imagine an investor is choosing how to divide each dollar between two investments, labelled 1 and 2. Investment 1 has a return with mean 1 and standard deviation 0.5, investment 2 has a return with mean 3 and standard deviation 2, and the correlation between the two returns is 0.6. The portfolio invests a fraction ω in investment 1 and a fraction 1 − ω in investment 2.

(a) Find formulas for the mean of the portfolio return, labelled µp, and the variance, labelled σ2p, as functions of ω.

(b) The investor would like a high return on average but also would like low variability (a measure of risk). Suppose the investor seeks the highest value of

µp − σ2p.

Find the value of ω that meets this objective.

6. Use the data set schoolexp from the Projects web page for this course. For a sample of 30 school districts in Indiana, it records the number of students (in thousands) and annual expenditures (in millions of dollars).

(a) Use Excel to produce and properly label a scatter plot of expenditures (on the vertical axis) plotted against the number of students (on the horizontal axis).

Next run a linear regression like this:

y = a + bx,

In Excel, go to the ‘data’ tab then to ‘data analysis” and ‘regression’.

(b) Report the coefficient on the x-variable (i.e. the number of students) and its standard error. Also report the R-square (R2) statistic.

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(c) Sections 2.3–2.4 of the textbook tell us how to interpret these statistics. Based on these statistics is there much evidence of a relationship between the number of students in a school district and its expenditures?

(d) A new school district is being established with an enrolment of 80 thousand students. Predict its annual expenditures.

7. Use the data set tornadoes, where the first two columns give the year from 1953 to 2008 and the number of tornadoes reported in the US each year.

(a) Use Excel to produce and properly label a scatter plot of the number of tornadoes (on the vertical axis) plotted against the year (on the horizontal axis).

(b) Run a linear regression of the number of tornadoes on the year. Report the coefficient on the x-variable (i.e. the year) and its standard error. Report the R-square (R2) statistic.

(c) Is there evidence of a statistical relationship?

(d) Predict the number of tornadoes in 2009.

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