Question

Suppose your friend Joanna is running for class president. The proportion of individuals in a population of students who will vote for Joanna on election day is 60%. You plan to conduct a poll of size n and report X, the number of individuals in your poll who plan to vote for Joanna. You also plan to compute ?̂=??, the proportion of individuals in your poll who plan to vote for Joanna.

a) Explain why X is a binomial random variable. What are the values of the binomial parameters n and p.

b) Show that E(?̂) = 0.6 and Var(?̂) = ?(1−?)? based on properties of the binomial distribution.

c) In a poll of size 10, what is the exact probability that ?̂ is greater than or equal to 0.5?

d) In a poll of size 20, what is the exact probability that ?̂ is greater than or equal to 0.5?

Hint: The r functions pbinom and/or dbinom can be helpful.

e) Using the statistical software R, sample n individuals from the distribution described above. For N = 10,000 iterations, compute ?̂=??, the proportion of individuals who plan to vote for Joanna for n = 50, n = 150, n = 500, n = 1000 individuals and plot a histogram of the sampling distribution of ?̂=??.

f) Estimate the mean and variance of the sampling distributions in (e) and compare to expected.

g) What do you notice about the shape of the sampling distributions in (e) as n gets large?

Answer 1