00:01
We have sample data.
00:02
The sample size n is 220.
00:06
P -hat, the sample proportion, is 0 .81.
00:10
We want a 90 % confidence interval for the population proportion.
00:15
So the formula for this is p -hat, the point estimate, plus and minus the margin of error, z root p -hat, 1 minus p -hat over n.
00:28
So we have n, we have p -hat, we just need z, and then we can calculate this.
00:34
So i'll explain where this comes from.
00:36
Initially, looking at this, you have a binomial experiment.
00:42
You have n independent trials, two outcomes, we'll call them success and failure, same probability p, whatever it is, of success on each trial.
00:52
We take a normal approximation to the binomial.
00:57
So we have this approximately normal curve, and like the binomial, it's a probability curve for x, the number of people in the sample who meet the criteria, the number of success states.
01:10
The mean of this, mu, is np.
01:13
The standard deviation, sigma, is root np, 1 minus p.
01:17
These are just the mean and standard deviation of the binomial distribution.
01:23
But i'm looking at p -hat.
01:24
So i take this distribution and its parameters and divide them all by n, turning it into a probability distribution for p -hat, the sample proportion.
01:36
Mu becomes mu of p -hat, and it's just p.
01:42
Sigma becomes sigma p -hat, also called the standard error, and it is p, 1 minus p over n, all in this square root sign...