>> Here's a quick tutorial on how to do hypothesis testing, when you're dealing with proportions,
specifically with one proportion.
I'm going to do this using [inaudible] method and the P value method.
So, we go through and read this, what we're given is that 70% of the population something,
that will always be true, in this case it's are bored by this video, but whatever, it
doesn't matter what it is, the key here is that this is about the population, this is
about everybody, so we're going to use the symbol P not for this.
We've been using P in the past for it, but we want to use P for something else here,
so we're going to call it P not.
In a sample of, so that will be key, you'll always be give population stuff and sample
stuff.
In a sample of 80 statistics students, who watch this video, only 50 were bored by it.
Your calculator uses X and N for these two things, this is the number of people in your
sample and this is the number of people that fit this criteria in that sample.
And, we want our conclusion to have 95% level of confidence, in other words, alpha is equal
to .05.
So, if you go through and do this problem, the way I'm going to always do these problems,
we're going to start out by stating the null and the alternative hypothesis, hypotheses
I supposes.
The null hypothesis will always be P relative to P not.
So, in this case, P not is equal to 70%, so I'll put .70 here, but both the null and alternative
hypotheses will always be P relative to P not, and the null will always be P equals
P not, whereas the alternative, you'll have to read the problem to figure out.
In this case it says, makes the video less boring, so we have less than here.
So, what really we have here is a left tail test, and again it's because, being in a statistic
class makes this less boring.
This P here is a population proportion, but specifically it's a population proportion
if the entire population were in a statistics class.
So, it's easy to confuse this with a sample proportion P hot, which I guess I can write
that off to the side, P hot which is X over N, in this case 5/8, 50 over 80.
This is similar to that, except this is population and this is sample.
Anyways, that will always be the first step, the second step for us will be to state the
shape, the center and the spread, so that we can later draw a picture; center then spread.
The shape is going to be approximately normal, that's where it always is in this class.
The reason why its approximately normal is the criteria we saw earlier was N times P
times 1 minus P had to be greater than or equal to 10.
That's not what it is here, because we don't know what P is, or we do know P not, we're
going to assume our null hypothesis is that P equals P not, so we can just use P not down
here.
So, really what we can do is say N is 80, P not is .70, one minus P not is point, I'll
write 1 minus .70, its .30, and if you do that multiplication, I do it in my head, you
get 16 point, 17 point, I'm going to go ahead and do it, I'm trying to show off here, 80
times .7 time .3, this is .3 in here, you get 16.8, might have said that or close, I
don't know.
Anyways, 16.8, sure enough, that's greater than or equal to 10, awesome, our shapes approximately
normal.
Center; we would like P to be the center, however we don't know P, but our null hypothesis
assumes that P equals P not, so we'll use P not for the center, which is .70.
And, our spread, sigma sub P hot is the symbol, but not a big deal that you know the symbol,
is, we have a formula for that, it's the square root of, see, in the past it was P times 1
minus P over N, that's not what it's going to be here, because, as I said, we don't know
P, but we assume P equals P not, we assume our null hypothesis is true, so we can make
those P nots, and then you can calculate this, if you really want to, sure, why not; .7,
.7 times .3 divided by, what was N, 80?
But that's not my spread, I need to take the square root of that, so I get square root
of answer and I get .05 ish, I don't know, 051, good enough.
So, I have the shape of the center in the spread, so now I can draw pictures and make
my conclusion.
Turns out that I'm going to draw different pictures, depending on which method of testing
I'm using.
I'm using the classical method, I'm going to draw a picture over here, and if I'm using
the P value method, I'll draw a picture over here.
Either way, I want something that's approximately normal.
[ Silence ]
There's a couple distributions that are supposed to be approximately normal, either way I want
the center to be .70.
[Silence] Okay, now I'll just deal with the classical for a minute.
If we're going to do this the classical way, what I got to do is set up my rejection regions,
I know this is a left tail test and that alpha is equal to .05, so I want a rejection region
over here on the left hand side equal to .05, and I can figure that out, I can use the inverse
norm function on my calculator and that will give me the Z score that has .05 below it,
so if I hit second and then distributions and then go down to inverse norm and put in
.05, you don't have to put in comma 0 comma 1, I do it just to be really clear that what
I'm getting here is negative 1.645, is really a Z score, which is why I write it down on
a different row here.
Z score has a center of 0, and this value goes here.
And so, now what I want to do is figure out my test statistic and if my test statistic
is in the rejection region, if it's less than negative 1.645, I'll just reject null hypothesis.
And, what's nice is my calculator has a function that will calculate the test statistic for
me, it's under tests, it's this 5th guy here, 1 prop Z test, prop standing for proportion,
1 because we only have one proportion, Z test; proportions are always Z test is because we're
only estimating one parameter, P, we're not estimating two parameters, like, for example,
sigma and MU.
Anyways, it's a one prop Z test, and we have all the information.
We know that P not was .70, X was 50, and N was 80, and our alternative hypothesis is
that P is less than P not, so it's the second guy here.
And so now, if I hit calculate, I get a test statistic of negative 1.46, close, hmm, I
wonder if I can move this over a little bit, what was the last thing I typed here, yeah,
I'm just going to slide this over a little bit so I have a little bit more room to write
here.
Here's that negative 1.645 from my critical value.
My test statistic is negative 1.5, I'm not going to carry very many decimal places because
I don't have a whole lot of room for it.
My test statistic, which corresponded with the 50 divided by 80 with P hot, 5/8 which
is .625, I'll just write 50 over 80 here so it's more apparent what it came from.
That test statistic is only over here, it's not in my rejection region, so what I would
say is because my test statistic.
[ Silence ]
Which, in this case, was negative 1.5, is not in the rejection region.
[ Silence ]
There is not sufficient evidence.
[ Silence ]
To reject a null hypothesis.
I'm not saying the null hypothesis is true, I'm just saying there's not enough evidence
to conclude that its false, so my test statistic is not in the rejection region, there's not
enough evidence to conclude that its false.
And that's the end of the problem if you were using the classical method, if you're using
the P value method, so now we're going to jump back, pretend you had done all this green
work, and then you come down here to draw the picture, it's a slightly different picture
if you're using the P value method, it turns out it's a little easier.
You put the exact same thing into your calculator, but now what you get is your test statistic
is the, you don't set up rejection regions, is in short what you do.
You still get your test statistic, it is negative 1.5 and that still corresponds with the P
hot, which is 50 over 80.
[Silence] Somewhere over here, exact same thing I did here, but now I don't make rejection
regions, all I say is that P value, which since this is a left tail test, is the area
to the left that I'm trying to shade in here, although I didn't do a great job of it, whatever,
I'm sure you don't care, is, and then it's this value right here, this .07.
This can get really confusing, .072, because you have P hot and P here, they look very
similar, they're not at all related, P hot at .625 is this fraction, 50 over 80.
P is a probability, it's the probability of getting a result this extreme, so it's the
value below here, but, anyways the .072 is what we need here.
So, when we're drawing our conclusion, we would say because my P value, which in this
case was .072, is greater than alpha, maybe I can do that, because the P value is greater
than alpha, there's not sufficient evidence to reject the null hypothesis.
So, P value, here's your picture, here's your conclusion, classical, here's your picture,
here's your conclusion.
Key thing to know is that you'll always get the same conclusion.
So, there's the two different methods dealing with a one proportion Z test.