00:01
We want to know the sample size necessary for a particular margin of error in a confidence interval for a population mean.
00:08
So if i did know my sample size n, i could go ahead, take my sample and find x -bar, the sample mean.
00:15
I could make a confidence interval around this to estimate the population mean.
00:19
The formula for such a confidence interval has the point estimates plus and minus the margin of error z sigma over root n.
00:29
I'm just going to focus on the error since we're told in the first part it has to be equal to 5 .3.
00:36
Now if i solve for n, i will know the sample size necessary to make this statement true.
00:42
So that's what we're going to do.
00:44
We have a couple of unknowns.
00:46
Sigma is the population standard deviation, 16 .2.
00:50
We get z from the level of confidence.
00:56
The central limit theorem tells me that as sample size increases, sample means become more and more normally distributed compared to the population.
01:04
If n is at least 30, they are approximately normally distributed.
01:09
So if i took every sample of size n, took the sample means and plotted them out, i'd have something approximately normal.
01:16
To make an interval, you put your point estimate in the centre and you form the interval around it.
01:20
In part a, that would contain 98 % of the sampling distribution.
01:26
That leaves 2 % in the tails, so each tail is 1%.
01:29
Z is the z -score to exclude these tails.
01:34
You might have a table of these.
01:35
If not, use the inverse normal function on software of your choice to get your critical value, 2 .326.
01:43
You might be asking, what if n isn't at least 30? well, you need this sampling distribution to be at least approximately normal for any of this to work...