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So the next step in our chi squared test of independence is going to be to get these two
expected frequencies -- this one and that one. So what you need in order to get that
is to recall what we computed before: that overall what normally happens is about a third of
all outings happen in the rain on our race course and about two-thirds, 0.67, of all
outings happen not in the rain. And then what you have to do is take the number without
accidents because where we're focused here is -- this is in the no accidents so we're
saying out of these guys, these 97 people here, how many of them would we expect to
fall into rain and how many will we expect to fall into no rain. So it's 97 is the sort
of base rate. So our expected frequencies if we're looking at no accident and yes, rain,
that's going to be 0.33 times 97 people. Pause, see if you can get the right answer on your
calculator. And it's going to be - -we're going to round just for mathematical simplicity,
I would not recommend you do this on tests, this is only to keep this real simple -- it's
going to be 32. Now see if you can come up with the expected frequency for no accident
and no rain. Pause the video and see if you can figure it out. Okay, if you did it right
about 0.67, about 67% of all outings are not in the rain. So the norm or default mode that
we should expect is 67% of the 97 people will be no accidents, no rain.
You're going to come up with about 65 to be expected frequency. So 65 is the
expected frequency here and 32 is the expected frequency here. Now the next challenge that
happens is you're going to actually do a chi squared test. The real challenge
here for most people is simply keeping track of what's what.
here for most people is simply keeping track of what's what.
The biggest mistake I see that will cause people points on tests for example
is people try and do stuff from memory and they don't write it all up and keep it on paper.
So I'm going to ask you to work with me here and label really conscientiously everything.
the first thing we're going to do is keep track of what cell we're talking about.
And the way I like to do that, I'm going to do accidents -- is there one or not, and rain -- is there one or not.
So we have one of the cells where there are accidents and there is rain. And in one of the cells we have accidents and no rain.
And in cell it's no accident, and there is rain. And in another cell it's no accident with no rain.
So these are our four conditions represented here.
So just transfer into the table.
I recommend, highly recommend tables because again it keeps you from getting things confused.
So looking back at your notes put all the expected frequencies here and they were -- look back at the table that you now have on paper --
they were 19, 20, 26, and 71.
This is the data we observed in our study. This is how many people fell basically into each of these conditions.
And then put the expected frequencies here that we just computed.
So what do we have -- a 13, a 26, a 32, and a 65 for expected frequencies.
Now we want to figure out how far what we found differs from what we expected by chance or what we expected if there's no relationship.
And we do that by making something similar to a deviation score -- it's observed minus expected.
So I'll do the first one with you and then I'd like you to do them for yourself and make sure you know how.
Observed minus expected, 19 minus 13 is 6.
Now pause the video, don't just sit here like a lump! I want you to really do these and see if you have it right,
because there are some common mistakes that people make and you want to make them now not while you're taking the test.
So get these three.
Okay, so hopefully you paused the video and when you took 20 minus 26 you got a NEGATIVE 6.
And when you did 26 minus 32 you got another negative 6.
and 71 minus 65 was 6.
Your accuracy check here is that these will sum to zero. And, they do.
And they always will if you've computed them right.
Okay, so that is accurate.
Then we're going to square these. When you square a number you take it times itself.
So take 6 times 6 is 36.
Now pause the video and do all the rest of these, don't assume you know how because people do make mistakes here.
Okay, pause it, do the calculations, and then un-pause it.
what you should get is negative 6 times negative 6 is positive 36.
And negative 6 times negative 6 is positive 36, 6 times 6 is 36.
Squares, which these are, are always positive.
So some of you will have put negative numbers and now you know not to do that.
A negative times a negative is a positive so when you're squaring negative numbers you come up with positives.
Now here's what I want you to NOT do, because this is a common mistake,
people will think they're doing an ANOVA, and they'll add these up to try to make a sum of squares right here.
Put an X right here, do NOT compute this.
This is NOT an ANOVA! It's actually constructed differently.
We still have one more step to go and that is: we're going to take these numbers and divide by the expected frequency.
So formulaically it would be the squares that we just made divided by expected frequencies.
And we happen to have those listed right here, don't we?
Right here are the expected frequencies. Isn't that handy?
Using the table is helpful in so many different ways.
So we're going to take 36 divided by 13, right?
And doing this on my calculator, 36 over 13 I get 2.77.
Now I'd like you to pause the video again and see if you can get the rest of these.