00:01
We want to form a confidence interval for the number of books people read in the previous year.
00:06
And we don't know how many subjects we need for our sample.
00:10
Imagine if we did know the sample size n.
00:12
We'd go ahead, take the sample, and find x bar, the sample mean.
00:17
From this we could form a confidence interval for the population mean, which is the average number of books people have read this year.
00:26
Formula is the point estimate, x bar, plus and minus the margin of error, z sigma over root n.
00:34
I'm just going to focus on this margin of error though.
00:38
So we want within six, within three, six again but with more confidence.
00:44
Okay, i'll set this as less than or equal to six for now.
00:48
And if i now solve for n, i'll know the sample size necessary to get the error within six books.
00:54
So i've got a couple of unknowns here.
00:57
Sigma is the population standard deviation.
01:00
We do have an estimate here based on an initial survey, but the standard deviation may be 15 to 7.
01:07
We'll use that for sigma.
01:09
Z is based on the level of confidence.
01:12
So for the first part we have 95 percent.
01:17
Okay, so the central limit theorem tells me that as sample size increases, sample means get more and more normally distributed.
01:25
Central limit theorem, if n is at least 30, the sample means are approximately normally distributed.
01:32
If i were to take every possible sample of size n, plot out the sample means i would get something approximately normal.
01:38
Again, assuming n is at least 30.
01:41
If not, then we need to assume the population is normally distributed because we need a normal distribution just to solve this question.
01:50
Anyway, you would be putting x bar in the middle and making an interval around it containing 95 percent of the area of the curve, or 95 percent confidence.
01:59
That leaves 5 percent of the tails, so each tail is 2 .5 percent.
02:04
Z is the z -score to exclude these tails...