00:01
Alright, so we have some data here and we want to make a confidence interval and a prediction interval.
00:06
And so the first part, a, asks us to find the standard deviation, y -hat, when x is for x equals 8.
00:19
Alright, so this is going to be given as this formula, s times the square root of 1 over n plus x minus x -bar squared all over the sum of all the x values minus the mean quantity squared.
00:40
So then this, i should say, this, i'll put a star here, this is where x is 8.
00:47
So this is going to be where x is 8.
00:50
And this x is for all the x's.
00:53
So we have n, n is going to be 5, that's good.
00:57
We need s.
00:58
S, i'm going to color code this, s is equal to the square root of the mean squared error.
01:07
And so for that, fortunately for us, we have, we ran a little regression table, this is what we get, and the regression table is right here.
01:21
And we read our little anova table, and the mean square is this, 76 .6 repeating, 76 and 2 thirds, so we take the square root of that to get our s value, and it's this.
01:35
That's what it is.
01:36
Alright, so that's our s, so we have that.
01:38
So we have x, n is 5, x minus the mean, so we need to get the mean, so we find the mean of the x's, so we get that.
01:45
This is just adding them all up, divide by 5 in this case.
01:50
You don't need the y's, but i have it there just in case.
01:53
Great, that's the mean, so 8 minus 11, take that value, square it, that's going to go here in the numerator.
02:00
And then we need this thing, the sum of the x minus x bar squareds.
02:04
So that, i'll do this in a separate color here.
02:09
This, we can find from the standard deviation of x.
02:13
So s sub x, the standard deviation of x is given as the square root of the sum of all the x values minus the mean of the sample, divided by n minus 1.
02:23
So we want to find, we want to isolate this term, the sum of x minus x bar squareds.
02:27
We want to isolate this.
02:28
So let's see if we square both sides.
02:32
That's going to get rid of that square root, so then we have the sum of x minus x bar squareds all over n minus 1.
02:39
So we multiply by n minus 1 on both sides, and we get the sum of squares of x.
02:46
And then, again, n is 5, so let's take that, multiply it by the square of the standard deviation, and we get this.
02:55
It's 180.
02:56
So now we have the denominator is 180.
02:58
So let's go ahead and fill this in.
03:02
So this is 180.
03:05
The numerator, 8 minus the mean, what was the mean? it was 11.
03:10
So that's going to be negative 3 squared.
03:12
That's going to be 9.
03:14
5, n is 5.
03:17
S is the square root of the mean square error.
03:19
It's this number, 8 .8, but just know it's that.
03:25
And that's going to give us the standard deviation right there.
03:28
That's it.
03:29
Bam.
03:30
Y sub hat on x is 8.
03:31
There we go.
03:32
So it's 4 .3779751.
03:35
Now, just so you know, i didn't round until the very end, so please be aware of that.
03:38
Like, i let the spreadsheet do all the calculations for us.
03:42
Now we're going to make a 95 % confidence interval.
03:44
So this is for part b.
03:48
95%.
03:51
Confidence interval.
03:52
So this is going to be y hat when x is 8.
04:04
All right, so there's that.
04:05
Plus minus t alpha over 2 with the degrees of freedom multiplied by sy hat when x is 8.
04:17
All right, so we just figured that out, too, didn't we? that's kind of nice.
04:20
So we have the alpha here for our confidence interval.
04:25
The alpha is going to be 0 .05.
04:29
The degrees of freedom.
04:30
Now it's important to note the degrees of freedom for a regression is n minus 2.
04:35
So 5 minus 2 is going to be 3.
04:38
And we can look up in a table or a spreadsheet, we get this number.
04:43
That's our t score.
04:44
3 .18.
04:50
We multiply it by this standard deviation of y hat.
04:54
X is 8.
04:55
So it's 4 .3 or 4 .8.
05:00
4 .38, we'll say.
05:02
Now we need this predicted value...