00:01
We're looking here at a normal distribution.
00:03
So i'll start by drawing it.
00:08
And the mean, mu, is 10 .2, and the standard deviation, sigma, is 0 .16.
00:18
For part a, we want the probability a randomly selected wedge of cheddar is greater than 10 .17.
00:26
So that's slightly below the mean.
00:28
We want the probability it's greater than that.
00:32
Okay, so with a normal distribution, we tend to use z scores, not raw data.
00:38
And to get the z score, we need x minus mu over sigma.
00:45
So here we have 10 .17 minus 10 .2 divided by 0 .16.
00:52
That's minus 0 .1875.
00:56
And we can turn this into a probability with either a z score table, a graphical calculator, or something like excel.
01:02
I'm going to use technology because it lets me use more decimal places in my z score.
01:09
And there are two functions.
01:11
If you use the standard, it gives you the area between your cutoff and the mean, which here is not the whole story.
01:17
You have to add 0 .5 for the area to the right of the mean.
01:22
If you use the cumulative function, it gives you the area to the left.
01:26
If you take that away from one, you'll be left with the area to the right.
01:30
So you can use either one.
01:31
I'll use the standard.
01:33
So i put in my z score.
01:34
And it gives me 0 .0744.
01:38
So i add 0 .5.
01:40
Let's get 0 .57744.
01:45
And that's part a.
01:48
Part b onwards, we are switching to a sample of size 15.
01:53
Okay, so we're looking at the mean weight of this sample.
01:58
Well, how are the mean weights of samples of size 15 distributed? they're also normal.
02:05
If the original distribution is normal, so are the sample means.
02:09
So if you plot every possible sample mean, you'll get a normal curve.
02:15
What's the mean of this curve? same as original, 10 .2.
02:20
There's no reason it should be higher or lower.
02:22
Some samples will have a higher mean weight, some will have a lower, but they'll average out 10 .2.
02:28
What's about the standard deviation? now, that does change.
02:32
The larger the sample, the narrower the distribution.
02:37
The less variation between the sample means.
02:40
And the way to calculate it is standard deviation of a sample mean is equal to the original standard deviation divided by root n.
02:50
So this is 0 .16 divided by root 15.
02:56
So let me find that.
02:58
0 .16 divided by root 15 is 0 .04 and 13.
03:06
So i'm going to keep it in this form to try and avoid any rounding errors.
03:15
Now for part b.
03:19
So approximately normal, it actually is normal because the original is normal.
03:26
You only have to worry about approximately normal if you're using the central limit theorem, which is where it's approximately normal by virtue of high sample size, even if the original distribution wasn't.
03:40
This actually is just normal.
03:42
So we've got mean 10 .2, standard deviation, 0 .0 points.
03:48
04, three decimal places.
03:56
So we want the probability the sample mean is less than 10 .17.
04:02
So it's still below the mean, and we want the area here.
04:06
So again, let's get the z score.
04:10
So 10 .17 minus 10 .2 divided by our standard deviation is now minus 0 .7 to 6, 3 decimal places...