- Imagine you are running a survey in Santa Barbara county with a colleague, with the
purpose of estimating the average annual income of an adult in the county. Instead of
surveying all the population, your colleague and you would only survey a sample size of
n individuals. You can assume through the whole problem that the random variable for
annual income, Wi
, is iid, with E(Wi) = µ and V ar(Wi) = σ
2
(a) (4 points) Write down the four assumptions required to make inferences for a population given a sample.
(b) (4 points) Propose an unbiased estimator for the average annual income in the
county. Show that it's unbiased.
(c) (4 points) Find the variance of your proposed estimator. Detail how the variance
changes as the sample size increases. Is this change a desirable feature? Explain.
(d) (3 points) Suppose that you and your colleague are in charge of surveying n
2
individuals each. For some reason, your colleague only surveyed people that live in
Montecito, where many celebrities live. What would this imply for your estimate
of the average annual income? - Suppose you conducted a poll on California residents in order to know if they think that
Climate Change is a serious threat to human life on earth or not. Let Xi
, for i = 1, …., n
be the answer for each surveyed selected in your poll, where:
Xi =
1 if they don't think it is a serious threat
0 if they do think it is a serious threat
Let's assume that the sample satises the 4 assumptions and that the probability that a
surveyed individual responds with a 1 is p (Meaning that the probability that a surveyed
individual responds that he/she doesn't think that Climate Change is a serious threat
is equal to p).
(a) (5 points) Suppose you want to test the hypothesis that 50% of California residents
think it is not a serious threat. Suppose we desire a 5% chance of mistakenly
rejecting this hypothesis when it is true. If you surveyed 250 people and 190 of
them answer that they do think it is a serious threat, what can you infer? Do
a hypothesis test (Make sure to write the null and alternative hypothesis, state the
test's statistic, write the decision rule and write one sentence with the result of the
test).
(b) (5 points) Use the same data of part a) to create a 95% condence interval for
1 − p, meaning the proportion of people who does think Climate Change is a
serious threat. Interpret the condence interval. (You can assume that the Central
Limit Theorem holds)
(c) (5 points) Forget about the 250 surveyed people in part a). Suppose now, that you
surveyed n people, yielding an estimation of pˆ = 0.53. Assuming you still want
a 5% test size, how big should the n be, so you can reject the null hypothesis of
p = 0.5?
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