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? 1 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 ...

1. 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

2. 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?

3. 1
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

4. 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

5. error. Also report the R-square (R2) statistic.
2
(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.
3

6. Lastname 3
Firstname Lastname
Instructor
Course
2 March 2016
Title
Please see the rubric for a complete explanation of how your
paper will be graded. In the first paragraph you summarize the
part of the article that you are analyzing in your paper. The
summary will briefly and clearly emphasize the particular issue
that you will be analyzing using the supply and demand model.
For example, the summary should start with a sentence such as,
“The author discusses rising US home prices and home sales
over the past two quarters.” Under no circumstance should your
summary exceed 100 words. For reference, this paragraph is
exactly 99 words in length.
For the remainder of the paper you will provide an economic
analysis of the article using the supply and demand model.
There are two ways you can approach the analysis. The first
approach is to explain a change that has occurred and the
second approach is to predict a change that may occur. You
should choose only one of these approaches. Whichever
approach you take, you must add something new to the article.
For example, if the article I’ve selected describes rising home
prices and explains that rising home prices are the result of
lower interest rates, I could not simply restate this as my own
analysis… because it isn’t. The analysis must be a new idea
from your own mind that extends the article. However, if I
thought the rise in home prices was due to another variable such

7. as rising income, then I could use this in my analysis.
As an example of explaining a change that has occurred, you
can use the supply and demand analysis to explain a change that
has already occurred in the price and quantity of a particular
good or service as discussed in the article you selected. For
example, the article may discuss rising home prices and home
sales. If you choose this option, your task is to identify the
cause of the change and explain how home prices (P*) and home
sales (Q*) rose using the supply and demand model. For
example, you might argue that rising home prices and home
sales are the result of rising incomes due to a recovering
economy. You would next explain how homes are a normal good
for which demand increases as incomes rise. In the graph below
you will illustrate the increase in demand and explain the
resulting increase in equilibrium price from P1 to P2 and
quantity of homes sold from Q1 to Q2.
Suppose that instead the article you chose describes rising
incomes. In this case you may choose the second approach, to
make a prediction about the future direction of home prices and
home sales (or any other market). In this case you would
explain, as before, that homes are a normal good and that an
increase in incomes is expected to increase demand resulting in
an increase in home prices from P1 to P2 and home sales from
Q1 to Q2. If you find yourself struggling on deciding what to do
with your article, it is often easiest to go with this approach.
Note that in either approach I will be looking for two key items
in your analysis. The first item is an explanation of the cause of
the shift in demand or supply. The second item is an
explanation of the effect of the change in demand or supply on
the equilibrium price and quantity in the market you are
analyzing. Here you will refer the reader to your graph below
and explain what the model predicts. For example, “As shown in
Figure 1 below, an increase in demand is illustrated by a
rightward shift in the demand curve from D1 to D2. This results
in an increase in the equilibrium new home price from P1 to P2
and an increase in equilibrium home sales from Q1 to Q2.

8. Therefore, the model offers a possible explanation for the
observed increases in home prices and home sales over the past
two quarters.”
As for the graph you will need to make changes to the template
provided below to suit your analysis. To do this, right click on
the figure, select Document Object, then click Open. This will
bring up a new Word document that you can edit. Changes will
automatically update in the template. If your analysis involves a
change in supply simply duplicate the supply curve and label
the new one S2, then delete D2. If you struggle with this please
catch me after class or come to office hours. Microsoft products
tend to make Macintosh products nauseous. If you have
formatting issues with your computer, please use a campus
computer to create your graph, save it as a picture file (e.g.
.png), then copy the image into your document.
For the citation, you will have only one source to cite. Some
examples are supplied to assist you.
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Figure 1
Works Cited
Lastname, Firstname. “Title of the Newspaper Article.” Title of
the Newspaper Date, edition: SectionPagenumber+.
“The Title of the Article.” Title of Magazine Date: page
number. Name of the Library Database: Name of the Service.
Name of the library with city, state abbreviation. Date of access
<URL>.
Lastname, Firstname. "Title of the Article." Website Title.
Publisher. Web.