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

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

  • 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 there we got 6.

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

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

  • We got 24.

  • 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.

  • And these are the scores on the 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.

  • And my question is, are these equal?

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

  • on how people perform on a test.

  • 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.

  • "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

  • is that if it doesn't make a difference,

  • 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.

  • How can we test this hypothesis?

  • 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.

  • So our 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--

  • Sum of Squares between

  • 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 SSW

  • 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

  • in the true population mean.

  • So if this number is really big,

  • it should tell us that there is a lower probability

  • that our null hypothesis is correct.

  • 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 let's actually calculate it.

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

  • and we had 2 degrees of freedom.

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

  • Also, 6. 6 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.

  • So our F statistic is going to be 12.

  • 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,

  • for our hypothesis test, is 10%.

  • 0.10 -- which means

  • that if we assume the null hypothesis, there is

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

  • of getting this F statistic,

  • then we will reject the null hypothesis.

  • So what we want to do is figure out a critical F statistic value,

  • that getting that extreme of a value or greater, is 10%

  • and if this is bigger than our critical F statistic value,

  • then we're going to reject the null hypothesis,

  • if it's less, we can't reject the null.

  • So I'm not going to go into a lot of the guts of the F statistic,

  • but we can already appreciate that each of these Sum of squares

  • has a Chi-squared distribution. "This" has a Chi-squared distribution,

  • and "this" has a different Chi-squared distribution

  • This is a Chi-squared distribution with 2 degrees of freedom,

  • this is a Chi-squared distribution with--And we haven't normalized it and all of that--

  • but roughly a Chi squared distribution with 6 degrees of freedom.

  • So the F distribution is actually the ratio of two Chi-squared distributions

  • And I got this--this is a screenshot from a professor's course at UCLA,

  • I hope they don't mind, I need to find us an F table for us to look into.

  • But this is what an F distribution looks like.

  • And obviously it's going to look different

  • depending on the df of the numerator and the denominator.

  • There's two df to think about,

  • the numerator degrees of freedom and the denominator degrees of freedom

  • With that said, let's calculate the critical F statistic,

  • for alpha is equal to 0.10,

  • and you're actually going to see different F tables for each different alpha,

  • where our numerator df is 2, and our denominator df is 6.

  • So this table that I got, this whole table is for an alpha of 10%

  • or 0.10, and our numerator df was 2 and our denominator

  • was 6. So our critical F value is 3.46.

  • So our critical F value is 3.46--this value right over here is 3.46

  • The value that we got based on our data is much larger than this,

  • WAY above it. It's going to have a very, very small p value.

  • The probability of getting something this extreme,

  • just by chance, assuming the null hypothesis,

  • is very low. It's way bigger than our critical F statistic with

  • a 10% significance level.

  • So because of that we can reject the null hypothesis.

  • Which leads us to believe, "you know what, there probably

  • IS a difference in the population means."

  • Which tells us there probably is a difference in performance

  • on an exam if you give them the different foods.

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

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