00:01
For this problem, we want to find a minimal sample size given some confidence interval information.
00:08
So we're told that we want to have less than a 0 .01 margin of error.
00:15
So let me write that the margin of error should be less than 0 .01.
00:20
We want a confidence level of 96%, so 96 % confidence interval.
00:27
And we have a point estimate for our proportion, which is 0 .62.
00:35
Okay, so what is this margin of error term? well, recall that if you want to construct a confidence interval for the population proportion, you take your point estimate p -hat, and you add or subtract the margin of error term, which is a z -score associated with this level of confidence, 96%, times the square root of p -hat times 1 minus p -hat over n, where n is our sample size.
01:03
So this term here, this is the margin of error, and we have a bound for this.
01:09
So using the bound for the margin of error, we can find a bound for n, and that will be our minimum sample size.
01:17
So in order to do this, we just need to find this z -score first, so i'll explain briefly how to do that.
01:22
So to find this z -score, imagine you have a standard normal distribution, looks something like this.
01:29
The mean zero is here in the middle, and the standard deviation is 1.
01:33
This z -score is such that the area between minus z sub c and plus z sub c is 96 % of the area under the curve, so that's 0 .96 from here to here.
01:46
And due to the symmetry of this situation, that means the remaining 0 .04 area is in the two tails equally distributed, so 0 .02 in each tail.
01:59
And this tells us that the area to the left of z sub c is 0 .98, which we can write down as a cumulative probability as follows.
02:11
Now, there are many ways to turn this cumulative probability into this z -score.
02:15
Tables, calculators, programs.
02:18
I'm going to use a ti -84.
02:21
It has the inverse normal function, it's called invnorm...