Pset #2
Due: October 3, 2023 (Tues)
Time: 11pm EST (Via Canvas)
Econ 10
Intro to Statistics
1.
Every year in March there is a standardized exam for people who want to be licensed sheep herders.
It happens that, with probability 0.45, a person will pass this exam. In the process of screening
people, it turns out that among those who passed the exam, 60% had taken college courses in
biology. It happens also that 30% of all those who take the exam had college courses in biology.
Find the probability that a person with college courses in biology will pass the exam.
2. Amazon.com maintains a list of all registered customers, along with their email addresses. During July, they
sent coupons to 20% of their customers. They recorded that 5% of their customers made purchases in July,
and 40% of all purchases were made with coupons. In this problem we will compute the proportion of
customers sent a coupon in July who made a purchase in that month. For simplicity, we will assume that
customers either make 0 or 1 purchases in July.
2a. Consider a random customer, and define two events:
Coupon = the customer received a coupon in July,
Purchase = the customer made a purchase in July.
Express all percentages given in the problem statement as probabilities or conditional probabilities of
these two events. Example: P(Coupon) = 0.20.
2b. Use Bayes’ rule to compute the proportion of customers sent a coupon in July who made a purchase that
month.
3. The following table shows the number of pairs of shoes reported and the genders of the 63 survey
respondents who answered the question “How many pairs of shoes do you own?”.
Pairs of Shoes
Gender
1–3
4–6
7–9
10–12
>12
Total
Female
Male
0
3
1
13
4
11
9
7
11
4
25
38
Total
3
14
15
16
15
63
3a. If we pick a random survey respondent out of these 63, and it turns out that the respondent happens to
be Female, what is the chance that the respondent will have 10-12 pairs of shoes.
3b. Answer the previous problem using the equation:
P(B|A) =
!(#∩ &)
!(#)
3c. If we pick a respondent who owns 4{6 pairs of shoes, what is the chance that the respondent is male?
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3d. Find the conditional probability P(> 12 pairs of shoes | Female).
3e. Find the conditional probability P(Female |> 12 pairs of shoes).
3f. Explain the different between (3d) and (3e).
4. Out of the 70 students enrolled in Math 13, 25 are female (36%) and 45 are male (64%). Suppose that we
randomly select two different students.
A = the 1st student picked is male
B = the second student picked is male.
4a. What is the probability that both students are male?
4b. What is the probability that both students are female?
4c. What is the probability that one of the students is male and one of the students is female?
5.
Let 𝑋 be a random variable describing the number of cups of coffee a randomly-chosen Dartmouth student
drinks in a week. Suppose that there is a 10% chance that the student has one cup of coffee, 30% chance
that the student has two cups of coffee, 40% chance that the student has 3 cups of coffee, and a 20%
chance that the student has four cups of coffee.
5a. Let 𝑝(𝑥) be the probability distribution function of 𝑋. Fill in the following table:
5b. Find 𝐸(𝑋), the expectation of 𝑋.
5c. What is the interpretation of the expectation of 𝑋?
5d. Find 𝑣𝑎𝑟(𝑋) and 𝑠𝑑(𝑋), the variance and standard deviation of 𝑋.
6.
Scratch ‘n Win.
You and three friends play a “scratch ‘n win” game at McDonald’s. You each have a card, and the cards say
“one out of three cards wins!” You decide to split whatever each of you wins among the four of you.
6a. Use the binomial distribution formula to calculate the probability distribution function for the number
of wins, that is, the probabilities that among the four of you there are 0, 1, 2, 3, or 4 winning cards.
The probability of winning with any one card is 1/3. This is the probability of “success” in a Bernoulli
trial. Since three friends and I are playing, there are four cards, or four trials. Here is the pdf:
Where 𝐵 = binomial random variable, number if winning cards
𝑛 = the number of trials (𝑛 = 4)
𝑝 = the probability of success (𝑝 = 1/3)
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6b. Using this probability distribution function, calculate the expected number of winning cards among
the four of you.
6c. Calculate the standard deviation of the number of winning cards.
6d. Calculate 𝐹(3).
6e. Most people seem to think that in a situation like this, one of you will surely win.
Why is this not true?
7. Stock analysis:
You are a stock analyst in charge of covering Acme Anvil Corp.
If they successfully close a deal with the Wyoming government, earnings next quarter will be high ($10
million). If they don’t close the deal, earnings will be low ($8 million).
You also know that their CEO is considering a job offer from another company. There is a 0.25 probability
that the CEO will take that job. If she takes it, there is a 0.5 probability that Acme will close the deal. If she
stays with Acme, on the other hand, there is a 0.8 probability that Acme will close the deal.
7a.
Fill out a tree diagram that describes this problem. (Hint: There are two sets of branches to the tree.
One corresponds to whether the CEO stays and the other corresponds to whether the deal is closed.
You can order the branches either way, but the easier approach is to make the second set of branches
match up with the conditional probabilities you are given in the problem.)
7b.
On the basis of your answer to (7a), give the probabilities that correspond to the following phrases:
7c.
i.
The probability that the deal is closed and the CEO leave
ii.
The probability that the deal is closed
iii.
The probability that the CEO stays conditional on the deal being closed
iv.
The probability that the firm makes $10million next quarter
Let the random variable C = 1 if the CEO stays, and C = 0 if she leaves.
Let the random variable E be Acme’s earnings.
Fill out the following chart illustrating:
fEC(e,c) à the bivariate pdf, 𝑓E(e) and 𝑓C(c) à marginal pdfs
Low E = $8m
High E = $10m
𝑓C(c)
C = 0 (Leave)
C = 1 (Stay)
𝑓E(e)
7d.
Are C and E independent? Why or Why not?
7e.
What is E(E|C=1)? What is E(E|C=0)?
Describe what these numbers (these conditional expectations) represent.
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8.
Suppose 𝑋 and 𝑌 are randome variables with 𝑣𝑎𝑟(𝑋) = 4, 𝑣𝑎𝑟(𝑌) = 3 and 𝑐𝑜𝑣(𝑋, 𝑌) = -2.
8a. Find 𝑣𝑎𝑟(𝑋 + 𝑌).
8b. Find 𝑣𝑎𝑟(2𝑋 + 5𝑌).
8c. Find 𝑣𝑎𝑟(3𝑋 − 𝑌).
9. Unemployment Insurance Claims Data:
9a. Who publishes the unemployment insurance claims data? What part of the US government?
What’s the url for the pdf released on April 20, 2023?
9b. What is it? How often is it released? When does it get released to the public? When does it get released
to the White House?
9c. What are the advantages of using the unemployment insurance claims data over the unemployment rate
for any analysis?
9d. What is the seasonally adjusted initial claims for the week ending April 15 (2023)? Was it an
increase/decrease from the week before?
9e. Which states experienced an increase of more than 1000?
9f. What is the relationship between the unemployment insurance claims data and the stock market
performance? Why does this data “move the market” (i.e. wallstreet)?
9g. How did the stock market react to the news Thursday morning? (Dow Jones Industrial Average?
NASDAQ index?)
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