00:01
All right, so we're looking at 20 different electric space heaters with different wattages.
00:06
And we want to compute the correlation between the wattage and the heating area.
00:12
So here's the data.
00:13
There's a lot of stuff here and i'll go through this piece by piece as we introduce questions.
00:18
So i put the data into heater, wattage area columns to help see the values a little more clearly and work with them more easily.
00:27
So i used the corel function on my spreadsheets.
00:33
It's corel.
00:36
And that is a nice way to generate the, you put your x list and your y list.
00:48
And it pops out the correlation coefficient, which is super nice.
00:53
And so we get 0 .0592.
00:57
And we want to go to how many decimals, four decimal places.
01:00
So 0 .05 .9.
01:04
Let me just double check.
01:06
0 .0592.
01:09
Yep, that's four decimal places.
01:11
There you.
01:11
And this tells us there's a direct relationship, meaning as the wattage goes up, the area goes up.
01:19
That's what this tells us.
01:21
The indirect if it would be indirect if it would be indirect, but this is direct.
01:33
We're going to test the hypothesis.
01:37
We're going to conduct a test of hypothesis, that is, to determine if it's reasonable that the coefficient is greater than zero.
01:45
The slope of this least squares regression line is, in fact, greater than zero at the alpha .05.
01:54
So what we're going to do is take our, we're given the hypotheses here, the row, well, they use row instead of r.
02:08
We'll use that just to be consistent because row.
02:11
The null is that row is less than equal to zero.
02:15
That's the null hypothesis.
02:16
The alternative, this is the thing we're testing, is row greater than zero.
02:21
And we're told to use the t test and reject h0 if t is greater than 1 .734.
02:28
So i have the t value here.
02:31
0 .25.
02:31
And the way you get that is the following.
02:34
T equals r times the square root of n minus 2 all over the same.
02:39
Square root of 1 minus r squared.
02:41
Well, i guess, like i said, we'll use row here.
02:48
So you do the appropriate substitutions.
02:50
Oh, we should say n is 20.
02:52
That's something we need to define here.
02:54
N is 20.
02:57
So you substitute end 20, right in the 20.
03:02
So you get 0 .0592 times the square of 18 divided by square root of 1 minus r value, or the row value squared.
03:13
And we get this t value.
03:15
So that means we fail to reject.
03:17
So we would fail to reject because the t value is 0 .2518, 9 to 2 decimals.
03:26
So t equals 0 .252.
03:35
Right? yeah, 0 .252 rounding to 3 decimal places.
03:42
Perfect.
03:42
And that means we fail to reject h0.
03:52
So that means it's not reasonable that the coefficients.
03:55
Grade in zero.
03:57
And i actually put this into a chart to help visualize the data a little more clearly.
04:05
Here are the wattages right here in the given areas on the y -axis.
04:12
And the blue is the data.
04:16
And so you can see it's kind of scattered.
04:18
There's not a huge upward linear trend.
04:21
But there's technically some positive relation to it, but statistically that's not significant.
04:30
That said, we're going to develop the regression equation for effective heating based on wattage.
04:36
And the way i did this, i have, we're going to make our white hat values equal a plus b x, where a is y bar minus b times x bar.
04:51
So we need b.
04:53
That means b need b first, so we get row times the sample.
04:59
Standard deviation of y, number sample standard deviation of x.
05:03
So back to this spreadsheet here, we find the mean of the x's, the means of the y's, and wattages are x areas of y, just to be clear.
05:17
Use this spreadsheet to find the sample standard deviations of each.
05:25
And then we substitute our values into the formulas, and we end up with the a value of 1303 .050 and the b value of 0 .03.
05:42
And we're rounding to three decimal places.
05:45
So put it all together, we get y hat is equal to 0 .013x plus 133 .050.
06:03
Let's see, what's the heating area that corresponds with 1 ,500? 100 watt heater.
06:11
So we take 1 ,500 substitute in for x.
06:15
So we do that.
06:26
And then we get this value, 152 .5.
06:48
And i guess we could, if we didn't round the regression equation to three decimal places, we could get more than four decimal places here.
07:02
But we did, so we're only given two.
07:05
So there we go.
07:07
What's the 95 % confidence interval of heating the area if the y is 1 ,500? so for this, this one, there's a few different calculations we can do to get this, and i'll show you where the different ways are.
07:27
We're just going to do one way just to make it easy on ourselves.
07:30
So our confidence interval is going to be y hat, our estimated value, which is that 152 .5.
07:40
5.
07:41
So that's easy enough, plus minus w.
07:44
Easy enough, right? that's great...
