00:01
Problem number 39 gives us a data set, and let's look at that data set.
00:04
This is with miles of railroad tracks.
00:08
So here's your x value, 15, 17, 38, et cetera, and then the y value there.
00:13
And the first part of this is to write a regression equation.
00:19
And so i'm going to use technology for that.
00:21
And so let's go to stat and then calc, and then we'll go down to that eighth option.
00:27
And there it is.
00:28
So we have negative 6 .763 plus 1 .755x.
00:35
So let's write that down.
00:38
Negative 6 .763 plus 1 .755x.
00:45
And then part b asked us to say whether it was a good fit or not or whether it predicted well.
00:53
And i mean, it seems to be pretty good.
00:55
So the r squared is 0 .713.
00:58
So that means we can explain 71 .3 % of the variation in y given the x, whatever those were, i can't remember.
01:05
But you could also graph this thing if you wanted.
01:08
So here's your regression line and your data values.
01:11
It seems to follow it pretty well.
01:12
So i would go ahead and say yes for this.
01:15
All right.
01:15
So yes.
01:16
All right.
01:16
Now let's find the 95 % confidence interval for x equals 30.
01:21
All right.
01:22
So the first thing we need to do is find what y hat star is.
01:27
And that's where you just plug in 30.
01:29
X so negative one negative 6 .763 plus 1 .755 times 30 so my y hat star my y star hat is 45 .887 so that's my point estimate my t alpha over two is the critical value so i can use a calculator for that too i'm going to go to inverse norm i'm not inverse norm inverse t sorry inverse t and the um area is 0 .025.
02:03
It's 95%, so that's an alpha of 0 .05, and then you cut it in half.
02:06
That's what the alpha over 2 means.
02:08
And the degrees of freedom here are going to be 5, since the end is 7.
02:15
Okay, so let's go ahead and paste.
02:17
Now, this is going to give us the negative, but it's positive and negative with confidence intervals.
02:22
So positive negative, 2 .5706, let's say.
02:26
So that's the t alpha over 2.
02:27
I should say positive negative, so 6.
02:32
And then the last thing, so let's say that's the y hat.
02:37
That's fine.
02:38
Okay.
02:38
And then the last thing we need is this little rascal here.
02:41
So here i did most of this in excel already.
02:44
So here's the data set.
02:46
And i needed x bar.
02:47
So i took the average of a1 to a7.
02:50
And then if you look at this formula here, it's x star, which that's that 30 minus the x bar squared.
02:57
So here, that numerator there is just going to be 1 because 30 minus the 29.
03:02
Is 1, and then the end is the sample size, so that's 7.
03:05
But then over here we have the summation of xi minus x bar squared.
03:09
So that's where i used excel.
03:11
I did the each data value, that's what the xi means.
03:15
So 15 minus the x bar, which was 29, and i squared it.
03:19
So i did that for all of them.
03:20
So that would be 17 minus 29 squared, you know, 38 minus 7 minus 29 squared, et cetera, etc...