00:01
So first of all, we are given a normal distribution, right? we are given a normal distribution.
00:06
We are told that the mean of that distribution is 47, right? this is the variable age, and we're told that the standard deviation is six, right? so the central limit theorem tells us something very important here, right? it tells us what the sample is, right? age is equal to x.
00:26
X bar is equal to our sample, right? our sample mean.
00:31
And the central limit theorem tells us that x bar is going to be normally distributed with the mean of the underlying population and sigma over root n, or n is the sample size.
00:43
So here, that is a normal distribution of 45 with 6 over 5, or i should say root 25, right? that's what the central limit theorem says, right? so here, if this is the distribution of x, the distribution of x bar is going to look way more like this, right? x bar.
01:06
Because when you take a sample, the pluses and minuses tend to cancel out.
01:10
So now i'm asking the probability of the probability that x bar is less than 47 .5, right? and that would be looking like all this area here, right? if this point here is 47 .5, we're thinking about all the probability mass to the left of that point.
01:30
So there's a whole bunch of ways to do this.
01:33
The old -fashioned way would be standardized to a normal distribution and use table, but that would be truly ancient.
01:45
The more modern approach is to use some piece of software, right? to use a computer to do this calculation for us...