00:01
All right, so here we have a nice little inference about a population proportion.
00:05
So we're given this statistic here that 47 ,000 american households, 32 .4 % own a cat.
00:14
So the answer to one is this a parameter or a statistic? it's a statistic because it comes from a sample, whereas a population, a parameter is from a population.
00:25
It's like a set number.
00:27
And that's what we're inferring.
00:29
We're trying to figure out what the true population.
00:31
Parameter is.
00:32
And so the symbol we use for this statistic is p -hat.
00:37
The other, the parameter would be p.
00:40
You might see this p -not, but for our statistic though, p -hat.
00:48
Describing words, the parameter of interest here.
00:50
In words, the parameter of interest is, it's the proportion of american households that own a cat.
01:09
Right, number three, we're going to theory -based significance test so we need a null and alternative hypothesis well first off we're connecting at the alpha of 0 .001 we will do the null so the null is the hypothesis in which we're kind of assuming we live in the thing we're we're testing against so we're we're testing for whether the sample data provides strong evidence that less than one -third of all of american households will catch that means that the true proportion would be greater than or equal to one third.
01:48
We're saying, no, is it actually less than a third? that's what we're testing against.
01:55
But this is what we're testing.
01:57
This is what we're testing against.
01:59
And we're seeing, is there evidence to suggest that it's less than a third? like this is the world, like one of my old staff professors would say, this is the world in which we live.
02:09
This is what we're seeing.
02:10
Or this is the believed world.
02:13
But is it actually this.
02:14
So we're going to do a little z test, z p hat, just given by p hat minus p over sigma.
02:28
So what we're going to do is we need these, we can kind of start filling this in.
02:42
0 .324 is our p hat minus 1 3 3rd, that's our p.
02:46
The sigma, this thing, is equal to the square root of p 1 minus p over n.
02:58
In this case, n is 47 ,000.
03:07
So we just plug all those values in.
03:12
So we get one -third times two -thirds all over 47 ,000 is that.
03:27
And we get 0 .00217.
03:32
Make that.
03:35
So 0 .0217 and that gives us a z value of negative 4 .29.
03:45
And so at this point what we're going to do is we're going to look up in our table.
03:49
So the probability that z is less than negative 4 .29.
03:54
To do a table lookup and that ends up being it's a very, very small number.
04:03
It's five zeros and then an eight.
04:10
So this is the probability of observing this proportion, 0 .324, if this thing were true.
04:20
That's what that means.
04:22
So that's a really small probability...