Economic statistics

Economics : Homework 5

Please provide written answers to Question 1 and R scripts for Questions 2 and 3.

Question 1 (60 points) Pollsters interview a simple random sample of n Texans about their preferences in the upcoming Senate race, asking “Will you vote for Cruz, Beto, or ‘Other’?”
Respondents choose one (and only one) of these options.
In the population, the true probability of voting for Cruz is pC, and the true probability of
voting for Beto is pB, leaving 1 ! pC ! pB as the probability of voting for “Other”. Think
of “Other” as a catch-all category that includes responses such as “I’m voting for someone
else” and “I’m not voting, period”.
Let the random variable Ci = 1 if respondent i says she will vote for Cruz, with Ci = 0
otherwise. Let the random variable Bi = 1 if respondent i says she’ll vote for Beto, with
Bi = 0 otherwise.
(a) Please fill in this 2 ⇥ 2 table of joint probabilities for respondent i:
Bi
0 1
Ci 0
1
(b) What are the expected values of random variables Bi and Ci? What are their population variances?
(c) What is the population covariance of Bi and Ci?
(d) What is the population variance of Bi ! Ci?
(e) Let µˆ B be the sample mean of B in a poll of n respondents, that is,
µˆ B = 1
n
nÂi=1
Bi,
and let µˆC be the sample mean of C in the poll.
1We (and the national media) are very interested in Dˆ , the difference of these sample
means,
Dˆ = µˆB ! µˆC
What is the expected value of Dˆ ? What is the population variance of Dˆ ?
(f) Suppose that the true pB = 0.48 and the true pc = 0.47. (This is a tight race!) Let the
sample size of the poll be n = 100. When the poll is conducted, what is the minimum
probability that the sample difference Dˆ will lie somewhere between !0.01 and 0.03?
Please show all key steps of your work. (Hint: Use one of the forms of Chebyschev’s
Inequality.)

Question 2 (20 points) (Based on Chapter 4 review question in Floyd [2010], page 127.) Using
R, find the following probabilities for the standard normal random variable Z:
(a) Pr(!1  Z  1)
(b) Pr(!2.16  Z  .55)
(c) Pr(Z $ !2.33)
(d) Pr(Z > 2.33)

Question 3 (20 points) (Based on Chapter 4 review question in Floyd [2010], page 128.) The population mean of a random variable Xi is µ = 40. The population variance is s2 = 81. Also,
Xi is normally distributed. Consider a sample of size n, which we denote by {X1, X2, . . . , Xn}.
We assume that all Xi variables have the same expected value and variance, and assume
that they are all uncorrelated.
Answer the following:
(a) What is the expected value and the variance of the sample mean µˆ when n = 36?
(b) Using R, find Pr(µˆ $ 41) when n = 36.
(c) Using R, find the probability that 38.5  µˆ  40.5 when n = 64.
2