If n=330 and p-hat=0.5, construct a 90% confidence interval

Answer 1

90% confidence implies alpha=0.1 so alpha/2=0.05. The z-score for this alpha is the z such that P(Z<z)=0.05 where Z is Normal(mean=0, variance=1) = Gaussian(0,1). From a calculator or z-table, we can see this is 1.645.

Now we have two options for the confidence interval:

One with the assumption that the sample variance can approximate the population variance (Central Limit Theorem): Confidence interval = p_hat +- z_score * sqrt(p_hat*(1-p_hat)/n) = 0.5 +- 1.645 * sqrt(0.5*0.5/330)

The other approach produces a larger interval that assumes the data does not provide information about the population variance:Confidence interval = p_hat +- z_score * sqrt(1/n) / 2 = 0.5 +- 1.645 * sqrt(1/330) / 3