1. Show your work and justify all steps. You will lose points if your reasoning is incomplete.

2. Submit your answers in the order in which they are asked in a single pdf document on

GauchoSpace.

3. Review your answers to ensure they can be read and understood by another person, especially if you handwrite them.

Problem 1. You’ve been hired to conduct a poll to assess whether a ballot proposition is likely

to pass.

(a) State the four assumptions necessary for your poll to provide useful insight. (4 points)

(b) Some media outlets have reported that supporters of the proposition may be ‘shy’ about

expressing their support. If true, which of the assumptions for polling would be invalid? If

true, would you be likely to overestimate or underestimate support for the proposition? (2

point)

(c) You have two assistants helping you conduct the poll. The three of you have agreed on the

number of voters to sample and the estimator. However, one assistant proposes sampling

with replacement. The other proposes sampling without replacement. Which proposal will

lead to a lower variance for our estimator? Explain (1 point)

(d) Assume there are N registered voters of which s plan to vote in favor of the ballot proposition. What is the probability that the first person you sample plans to vote in favor of the

ballot proposition? (2 points)

(e) Again, assume there are N registered voters of which s plan to vote in favor of the ballot

proposition. You plan to sample without replacement. What is the probability that the

second person you sample plans to vote in favor of the ballot proposition? Are Y1 and Y2

independent? (2 points)

Problem 2. New Caledonia is a French territory located in the Pacific Ocean. With your

recent work on the state ballot behind you, you’ve been hired to conduct a poll to determine

support for independence. Individuals in New Caledonia are in favor or against independence.

Let yk represent the response of individual k from the population. Let yk be coded as follows,

yk =

0, if individual k is against independence

1, if individual k supports independence

Let p represent the proportion of the population in favor of independence. You plan to collect

a random sample (with replacement) of size n, (Y1, Y2, . . . , Yn).

(a) Is yk a random variable? Explain. (1 point)

(b) One of your colleagues proposes the following estimator ˆp =

1

n

Pn

i=1 Yi

. Show that this

estimator is unbiased. (Show and explain your steps). (3 points)

(c) From class we know that variance of Y1 is p(1 − p). Derive the variance of the estimator

from part b. (3 points)

(d) Can you compute the variance with your sample? If not, explain and propose an alternative

that you can compute. (1 point)

(e) Your client wishes to have a margin of error of 5 percent. Using 95 percent confidence and

an assumed value of p = 0.5, calculate the sample size required for the margin of error. (1

point)

(f) Unfortunately, you have already collected a sample of 200 responses. It’s now one month

later, but you proceed by collecting a new sample of 185 responses. Your colleague proposes

combining the samples to achieve your desired margin or error. Can you use the combined

sample to obtain an unbiased estimate of the level of support on the day you conducted the

first poll? Explain. (1 point)

Problem 3. After much deliberation about how to conduct your poll to determine support

for independence in New Caledonia, you decide to poll 2000 individuals. Of these, 860 respond

in favor of independence and 1140 respond against. Using the estimator from the previous

question you find that ˆp2000 = 0.43. France will grant New Caledonia independence if more

than 50 percent of the population vote in favor.

(a) Using your survey results, compute a 99 percent confidence interval for p, the proportion

of the population in favor of independence. (3 points)

(b) Using the confidence interval you computed above, explain if you think it is likely that the

population of New Caledonia will vote in favor of independence. (3 points)

(c) In October 2020 a vote was held and 46.7 percent of the population voted against independence. Your client asks you how it is that this was outside your confidence interval? Explain

why the population parameter might fall outside the interval you calculated. Assume full

voter turnout and that the four assumptions for polling were valid for your poll. (3 points)

Page 2

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