Second regression example Second Regression Example
Second regression example
- Let's find the equation for the regression line
- that best fits this.
- Where the fit minimizes the squared distance to each of
- the points.
- And then let's actually calculate how good of a fit it
- is using an r squared.
- And we might have to do that in the next video,
- depending on time.
- So just as a reminder, the line is going to have the
- equation y is equal mx plus b.
- And we've shown ourselves that the slope of this line-- the
- one that best minimizes the squared distance to each of
- those points-- is going to be the mean of the xy's minus the
- mean of x times the mean of y.
- All of that over the mean of the x's squared, or the mean
- of the x squareds, minus the means of the x's squared.
- So one way to memorize it, I guess, is the first terms have
- the mean of the combined things.
- You're multiplying x times itself first, then meaning.
- You're multiplying x times y, times each
- other first, then meaning.
- And then the second terms, you're finding the means of
- the individual components and then multiplying.
- Mean of x, times mean of y, mean of x times mean of x.
- So hopefully maybe that helps.
- Maybe it doesn't.
- But we can calculate the slope.
- And then the y intercept, b, is just going to be equal to
- the mean of y times whatever we calculate here for m, times
- the mean of x.
- And we can do that because we know that the point mean of x
- comma mean of y is going to be on this regression live.
- So what's calculate them.
- And you'll see, in the last example we did three points.
- We only have four points here.
- But the computations get more and more intense.
- You can imagine what would happen if you had 10 or 20 or
- 100 points.
- You pretty much have to use a calculator at that point.
- Or computer, even better.
- Or a spreadsheet.
- So let's calculate m.
- And to do that, let's calculate the components.
- So the mean of x-- the mean of the x's-- is going to be equal
- to, this x is negative 2, plus negative 1, plus 1, plus 4.
- All of that over, we have four x data points.
- These two guys cancel out.
- Negative 2 plus 4 is 2.
- 2 over 4 is equal to 1/2.
- Now let's do the mean of the y's.
- We have negative 3, we have a negative 1.
- And then we have a 2, and then we have a 3.
- And once again, we have four data points.
- That guy and that guy cancel out.
- Negative 1 plus 2 is 1.
- So this is equal to 1/4.
- Now let's figure out the mean of the xy's.
- So x times y, the mean of that.
- So over here we have negative 2 times negative 3.
- Negative 2 times negative 3 is positive 6.
- Plus negative 1 times negative 1 is positive 1.
- Plus 1 times 2 is 2.
- Plus 4 times 3 is 12.
- And we have four of these points.
- And what is this?
- This is 6 plus 1 is 7.
- 7 plus 2 is 9.
- 9 plus 12 is 21 over 4.
- This is equal to 21/4.
- And then finally, we want-- I'll do this in a new color--
- the mean of the x's squared.
- And so that is going to be equal to-- negative 2 squared
- is positive 4.
- Plus negative 1 squared is positive 1.
- Plus 1 squared is 1.
- Plus for 4 squared is 16.
- All of that over 4.
- 4 plus 2 is 6 plus 16 is 22 over 4.
- So 22/4 is the same thing as 11/2.
- So now we're now ready to calculate the actual slope.
- Let me do it over here.
- Well actually, let me do it over here.
- I want to be able look at everything we've done.
- So this is going to be equal to, in this case, it's going
- to be the mean of the xy's, which is 21/4.
- Minus the product of the mean of x, which is 1/2.
- Times the mean of the y's, which is 1/4.
- And then all of that over the mean of the x
- squareds, which is 11/2.
- So we did that.
- Minus the mean of the x's squared.
- The mean of the x's, once again, is 1/2.
- And so what is this equal to?
- I'm just going to go straight to the calculator.
- I could deal with the fractions, but this isn't a
- review of adding and subtracting
- and multiplying fractions.
- Let's just go straight to the calculator.
- Actually, let me simplify it before.
- It's just too tempting to simplify.
- Let me copy and paste it.
- Let's go down here to calculate it.
- And so this is going to be-- maybe I should have used the
- calculator, but it's too tempting.
- So what's this on top?
- On top, we have 21/4 minus 1/2 times 1/4 is minus 1/8.
- All of that over 11/2 minus 1/2 squared, which is 1/4.
- Now, one way to simplify this right from the get go is
- multiply the numerator and the denominator by 8.
- And that's just to get rid of all these fractions.
- So 21/4 times 8 is going to be the same thing is 21 times 2,
- which is equal to 42.
- Minus 1/8 times 8.
- We have to, of course, distribute the eights.
- So it's going to be minus 1.
- All of that over, 8 times 11/2 is going to be 11 times 4,
- which is 44.
- And then 8 times 1/4 is 2, so it's minus 2.
- So 42 minus 1 is 41.
- And then 44 minus 2 is 42.
- So the slope is 41/42.
- So a little bit less than a slope of one.
- 42/42 would be exactly 1.
- So our regression slope is a little bit less than 1.
- And then our regression y-intercept, b, is going to be
- equal to the mean of the y.
- So 1/4, minus our slope, minus 41/42, times the mean of the
- x's, so times 1/2.
- And so this is going to be equal to 1/4 minus 41/84,
- which is equal to-- let me just find a common
- So let's go over 84.
- So what's 1/4 of 84?
- 1/4 of 80 is 20.
- So this is 21.
- 21 times 4 is 84.
- This is 1/4 of 84.
- Yep, that's right.
- So it's going to be 21 minus 41 over 84, which is equal to
- negative 20.
- Negative 20 over 84, which is the same thing, they're both
- divisible by 4, the numerator divided by 4 is
- negative 5, over 21.
- So our regression line is going to be y is equal to
- 41/42 x minus 5/21.
- And 5/21 is a little bit less than 1/4.
- 5/20 would be 1/4.
- We made the denominator a little bit bigger, so it's
- going to be a little bit less than negative 1/4.
- So our y-intercept is going to be a little bit less than
- negative 1/4.
- And then we're going to have a slope a little
- bit less than 1.
- So our line is going to look something like this.
- If I were able to actually draw a straight line, it would
- look something like that over there.
- So I'm going to leave you there in this video.
- In the next video, we're actually going to calculate
- the r squared for this line.
- How good of a fit is it?
- How much of the total variation in the y values can
- be explained by the variation in the x values, or by the
- line itself?
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
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This is great, I finally understand quadratic functions!
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