Question on Central Limit Theorem

My question is:

A country is about to hold a referendum about joining the European Union. We carry out a survey of a random sample of adult citizens of the country. In the sample, n respondents say that they plan to vote in the referendum. We then ask these respondents whether they plan to vote Yes or No. Define X = 1 if such a person plans to vote Yes, and X = 0 if No.

Suppose that in the whole population 45% of those people who plan to vote are currently planning to vote Yes.

a) Let the sample mean = (Sigma Xi)/ n denote the proportion of the n voters in the sample who plan to vote Yes. What is the central limit theorem approximation of the sampling distribution of the sample mean here?

b) If there are n=50 likely voters in the sample, what is the probability that the sample mean is greater than 0.5?

Any help would be greatly appreciated :)

Re: Question on Central Limit Theorem

Hey sakuraxkisu.

Hint for a: What does the central limit theorem state about the distribution of the mean? Also, what is the mean of a binomial distribution? What is the variance of a binomial distribution? How do you use the mean and variance (both estimated) in terms of the distribution of a sample mean for CLT?

Re: Question on Central Limit Theorem

Hey,

Is X bar ~ N(mu, sigma^2/n)? And for the binomial distribution, the mean is np for parameter p, and the variance is np(1-p), so would the distribution of the sample mean be X bar ~ N(np, p(1-p)) ?

Would you also be able to help me with (b)? Thanks :)

Re: Question on Central Limit Theorem

Yes that is correct for large enough n and this is known as the Central Limit Theorem.

For the binomial the answer is that the distribution of the sample mean of a binomial is p(1-p)/n and not p(1-p).

The reason is that the variance of one Bernoulli sample is p(1-p) and sigma^2/n is p(1-p)/n.

For b) you simply calculate P(X_bar > 0.5) and since you know X_bar has a normal distribution with large enough samples (good approximation anyway by CLT) then you standardize your statistic and look at the normal tables for P(Z > z) = 1 - P(Z < z) for any z.

Note that if X_bar is N(mu,sigma^2) then Z = (X - mu)/sigma ~ N(0,1) = Z

Re: Question on Central Limit Theorem

I'm still a bit confused about (b) - what would the value of the parameter p be? Because when I try to standardize it, I get:

P(Z > (0.5- 50p)/(sqrt(p(1-p)/50))), and then I'm stuck.

Re: Question on Central Limit Theorem

Hint: What does the 45% figure tell you about the value of p?

Re: Question on Central Limit Theorem

Oh, I feel so silly now! Thank you :)