00:02
So this problem gives us this table with the national average pupil teacher ratio in public schools over years from 1990 to 2010.
00:21
So the first part of this problem asks us to find the equation for the lee squares line for this data.
00:29
And it asks us to use x as the number of years since 1990 and why the average numbers of people's teacher that's this ratio so to find the this equation it looks something like this it's a line equation and so the first thing we need to do is build this table.
01:02
So here, these first two columns is the data that we're given.
01:13
Here, this represents the years, this is the ratio, and we're doing x, the way it was suggested.
01:23
So, year's, number of years, since 1990.
01:27
Apart from these two, we need to find, for each entry, we need to find who x times y is.
01:35
That's this column, x squared, and y squared.
01:39
And on this last row, we have the sum for each column.
01:49
So this is the sum of all the xs, and so on.
01:53
So these values over here, these are the ones we're going to use to find this equation.
02:01
This equation.
02:04
Since we have nine different points or entries, n is going to be six, so this is all we need, and the definition for m for the value that m and b is it's.
02:25
So these are the definitions.
02:28
We just going to plug in the values that we found on this table.
02:34
So first we're going to do m and we set it 6.
02:38
The sum of xy is 946 .8.
02:50
And we subtract the sum of x, that's 60 times the sum of y, that's 97 .7.
03:11
And this goes over and that's six, and now we have n times the sum of the squares, and that's this column here, minus the square of the sum.
03:28
So this number is, they're talking about this number.
03:34
So first is the sum of the squares, that's 880 minus the sum of x is 60, and this number is squared.
03:52
So now that we have this expression, it's all constants, we can plug that into calculators, and we would get that the slope for this line is minus 0 .1.
04:09
So this is the value that goes here.
04:13
Now for b, we do the same thing.
04:16
We just plug in the values that we found in this table, starting with sum of y, that is 97 .7 minus m, that is this m that we found here, minus 0 .1 times the sum of x.
04:46
Again, that is 60, and all of this is over n that has value 6.
04:55
All right, so again, we plug this into the calculator, and we get that this is approximately 17 .4.
05:06
So this is the second value that we needed to know here to find the least squares line.
05:13
And now we can write it out explicitly.
05:16
These values, m is minus 0 .1 times x plus b, that we found, to be 18, sorry, 17 point.
05:36
So this equation is the least squares line.
05:41
And that is what we're asked to do in this first part.
05:45
So for the next part, we're asked to use the answer for part a, the equation, to predict the pupil teacher ratio in 2020.
06:02
So what this is asking us, right so this is what we found in part a and this is the relation between the year and the ratio and this this year means that x we're assigning x the value of 30 because that's we're talking about years from 1990 and they're asking us to predict this ratio so we're asked to find the corresponding y value.
06:40
To do that since we already found this on part a what we're going to do is we're going to substitute this value here in x.
06:55
This is part of the equation...