Name: ANOVA 3: Hypothesis test with F-statistic | Probability and Statistics | Khan Academy
Uploaded: 2016-11-19T19:11:54.000Z
Duration: 10 min 14 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

In the last couple of videos we first figured out the TOTAL variation in these 9 data points right here

Reflexive pronouns

and we got 30, that's our Total Sum of Squares. Then we asked ourselves,

Transition words & phrases

how much of that variation is due to variation WITHIN each of these groups, versus variation BETWEEN the groups themselves?

So, for the variation within the groups we have our Sum of Squares within.

And then the balance of this, 30, the balance of this variation,

came from variation between the groups, and we calculated it,

What I want to do in this video, is actually use this type of information,

essentially these statistics we've calculated, to do some inferential statistics,

to come to some time of conclusion, or maybe not to come to some type of conclusion.

What I want to do is to put some context around these groups.

We've been dealing with them abstractly right now, but you can imagine

these are the results of some type of experiment.

Let's say that I gave 3 different types of pills or 3 different types of food to people taking a test.

So this is food 1, food 2, and then this over here is food 3.

And I want to figure out if the type of food people take going into the test really affect their scores?

If you look at these means, it looks like they perform best in group 3, than in group 2 or 1.

But is that difference purely random? Random chance?

Or can I be pretty confident that it's due to actual differences

in the population means, of all of the people who would ever take food 3 vs food 2 vs food 1?

So, my question here is, are the means and the true population means the same?

This is a sample mean based on 3 samples. But if I knew the true population means--

So my question is: Is the mean of the population of people taking Food 1 equal to the mean of Food 2?

Obviously I'll never be able to give that food to every human being that could

ever live and then make them all take an exam.

But there is some true mean there, it's just not really measurable.

So my question is "this" equal to "this" equal to the mean 3, the true population of mean 3.

Because if they're not equal, that means that the type of food given does have some type of impact

So let's do a little hypothesis test here. Let's say that my null hypothesis

is that the means are the same. Food doesn't make a difference.

and that my Alternate hypothesis is that it does. "It does."

and the way of thinking about this quantitatively

the true population means of the groups will be the same.

The true population mean of the group that took food 1 will be the same

as the group that took food 2, which will be the same as the group that took food 3.

If our alternate hypothesis is correct, then these means will not be all the same.

So we're going to assume the null hypothesis, which is

what we always do when we are hypothesis testing,

we're going to assume our null hypothesis.

And then essentially figure out, what are the chances

of getting a certain statistic this extreme?

And I haven't even defined what that statistic is.

So we're going to define--we're going to assume our null hypothesis,

and then we're going to come up with a statistic called the F statistic.

which has an F distribution--and we won't go real deep into the details of

the F distribution. But you can already start to think of it

as the ratio of two Chi-squared distributions that may or may not have different degrees of freedom.

Our F statistic is going to be the ratio of our Sum of Squares between the samples--

divided by, our degrees of freedom between

and this is sometimes called the mean squares between, MSB,

that, divided by the Sum of Squares within,

so that's what I had done up here, the SSW in blue,

divided by the degrees of freedom of the SSwithin, and that was

m (n-1). Now let's just think about what this is doing right here.

If this number, the numerator, is much larger than the denominator,

then that tells us that the variation in this data is due mostly

to the differences between the actual means

and its due less to the variation within the means.

That's if this numerator is much bigger than this denominator over here.

So that should make us believe that there is a difference

it should tell us that there is a lower probability

If this number is really small and our denominator is larger,

that means that our variation within each sample,

makes up more of the total variation than our variation between

the samples. So that means that our variation

within each of these samples is a bigger percentage of the total variation

versus the variation between the samples.

So that would make us believe that "hey! ya know... any difference

we see between the means is probably just random."

And that would make it a little harder to reject the null.

So in this case, our SSbetween, we calculated over here, was 24.

And our SSwithin was 6 and we had how many degrees of freedom?

So this is going to be 24/2 which is 12, divided by 1.

Our F statistic that we've calculated is going to be 12.

F stands for Fischer who is the biologist and statistician who came up with this.

We're going to see that this is a pretty high number.

Now, one thing I forgot to mention, with any hypothesis test,

we're going to need some type of significance level.

So let's say the significance level that we care about,

that if we assume the null hypothesis, there is

less than a 10% chance of getting the result we got,