PL9fwy3NUQKwaIc7KWc8buW344941GStQL/D4aO26sEGrc.srt

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In general, the regression equation is given by

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this equation. Y represents the dependent variable

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for each observation I. Beta 0 is called

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population Y intercept. Beta 1 is the population

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slope coefficient. Xi is the independent variable

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for each observation, I. Epsilon I is the random

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error term. Beta 0 plus beta 1 X is called

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linear component. While Y and I are random error

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components. So, the regression equation mainly has

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two components. One is linear and the other is

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random. In general, the expected value for this

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error term is zero. So, for the predicted

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equation, later we will see that Y hat equals B

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zero plus B one X. This term will be ignored

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because the expected value for the epsilon equals

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zero.

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So again linear component B0 plus B1 X I and the

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random component is the epsilon term.

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So if we have X and Y axis, this segment is called

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Y intercept which is B0. The change in y divided

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by change in x is called the slope. Epsilon i is

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the difference between the observed value of y

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minus the expected value or the predicted value.

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The observed is the actual value. So actual minus

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predicted, the difference between these two values

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is called the epsilon. So epsilon i is the

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difference between the observed value of y for x,

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minus the predicted or the estimated value of Y

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for XR. So this difference actually is called the

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error term. So the error is just observed minus

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predicted.

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The estimated regression equation is given by Y

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hat equals V0 plus V1X. As I mentioned before, the

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epsilon term is cancelled because the expected

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value for the epsilon equals zero. Here, we have y

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hat instead of y because this one is called the

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estimated or the predicted value for y for the

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observation i. For example, b zero is the estimated

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value of the regression intercept, or is called y

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intercept. B one is the estimate of the regression of

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the slope, so this is the estimated slope b1 xi.

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Again, is the independent variable, so x1 it means

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the value of the independent variable for

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observation number one. Now this equation is

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called linear regression equation or regression

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model. It's a straight line because here, we are

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assuming that the relationship between x and y is

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linear. It could be non-linear, but we are

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focusing here on just linear regression. Now, the

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values for B0 and B1 are given by these equations,

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B1 equals RSY divided by SX. So, in order to

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determine the values of B0 and B1, we have to know

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first the value of R, the correlation coefficient.

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Sx and Sy, standard deviations of x and y, as well

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as the means of x and y.

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B1 equals R times Sy divided by Sx. B0 is just y

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bar minus b1 x bar, where Sx and Sy are the

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standard deviations of x and y. So this, how can

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we compute the values of B0 and B1? Now the

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question is, what's our interpretation about B0

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and B1? And B0, as we mentioned before, is the Y

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or the estimated mean value of Y when the value X

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is 0.

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So if X is 0, then Y hat equals B0. That means B0

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is the estimated mean value of Y when the value of

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X equals 0. B1, which is called the estimated

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change in the mean value of Y as a result of one

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unit change in X. That means the sign of B1,

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the direction of the relationship between X and Y.

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So the sine of B1 tells us the exact direction. It

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could be positive if the sine of B1 is positive, or

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negative. On the other side. So that's the meaning

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of B0 and B1. Now the first thing we have to do in

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order to determine if there exists linear

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relationship between X and Y, we have to draw

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a scatter plot, Y versus X. In this specific

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example, X is the square feet, size of the house

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is measured by square feet, and house selling

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price in thousand dollars. So we have to draw Y

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versus X. So house price versus size of the house.

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Now, by looking carefully at this scatter plot,

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even if it's a small sample size, but you can see

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that there exists a positive relationship between

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house price and size of the house. The points

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maybe they are close, a little bit to the straight

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line, it means there exists, maybe a strong

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relationship between X and Y. But you can tell the

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exact strength of the relationship by using the

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value of R. But here we can tell that there exists

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a positive relationship, and that relation could be

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strong.

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Now, simple calculations will give B1 and B0.

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Suppose we know the values of R, Sy, and Sx. R, if

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you remember last time, R was 0.762. It's moderate

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relationship between X and Y. Sy and Sx, 60

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divided by 4 is 117. That will give 0.109. So B0,

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in this case, is 0.10977. B1

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B0 equals Y bar minus B1 X bar. B1 is computed in

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the previous step, so plug that value here. In

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addition, we know the values of X bar and Y bar.

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Simple calculation will give the value of B0,

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which is about 98.25. After computing the values

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of B0 and B1, we can state the regression equation

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by house price. The estimated value of house

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price, hat in this equation means the estimated or

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the predicted value of the house price. Equals b0

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which is 98 plus b1, which is 0.10977 times square

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feet. Now here, by using this equation, we can

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tell number one, the direction of the relationship

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between x and y, how surprised and its size. Since

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the sign is positive, it means there exists

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a positive association or relationship between

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these two variables, number one. Number two, we

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can interpret carefully the meaning of the

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intercept. Now, as we mentioned before, y hat

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equals b zero only if x equals zero. Now, there is

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no sense about square feet of zero because we

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don't have a size of a house to be zero. But the

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slope here is 0.109; it has sense because, as the

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size of the house increases by one unit, it's

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selling price increased by this amount, 0.109. But

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here, you have to be careful to multiply this value

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by a thousand because the data is given in

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thousand dollars for Y. So, here, as the size of the

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house increased by one unit, by one foot, one square

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foot, its selling price increases by this amount, 0

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.10977. It should be multiplied by a thousand, so

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around $109.77. So that means, extra one square

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foot for the size of the house, it cost you around

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$100 or $110. So, that's the meaning of B1 and the

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sign actually of the slope. In addition to that,

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we can make some predictions about house price for

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any given value of the size of the house. That

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means, if you know that the house size equals 2,000

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square feet, so just plug this value here, and

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simple calculation will give the predicted value

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of the selling price of a house. That's the whole

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story for the simple linear regression. In other

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words, we have this equation, so the

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interpretation of B0 again. B0 is the estimated

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mean value of Y when the value of X is 0. That

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means if X is 0, in this range of the observed X

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values. That's the meaning of the B0. But again,

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because a house cannot have a square footage of

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zero, so B0 has no practical application.

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On the other hand, the interpretation for B1, B1

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equals 0.10977, that means B1 again estimates the

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change in the mean value of Y as a result of one

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unit increase in X. In other words, since B1

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equals 0.10977, that tells us that the mean value

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of a house increases by this amount, multiplied by

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1,000 on average for each additional one square

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foot of size. So that's the exact interpretation

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about P0 and P1. For the prediction, as I

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mentioned, since we have this equation, and our

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goal is to predict the price for a house with 2

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,000 square feet, just plug this value here.

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Multiply this value by 0.1098, then add the result

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to 98.25. This will give 317.85. This value should be

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multiplied by 1000, so the predicted price for a

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house with 2,000 square feet is around 317,850

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dollars. That's for making the prediction for

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selling a price. The last section in chapter 12

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talks about the coefficient of determination R

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squared. The definition for the coefficient of

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determination is the portion of the total

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variation in the dependent variable that is

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explained by the variation in the independent

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variable. Since we have two variables X and Y.

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And the question is, what's the portion of the

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total variation that can be explained by X? So, the

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question is, what's the portion of the total

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variation in Y that is explained already by the

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variation in X? For example, suppose R² is 90%, 0

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.90. That means 90% in the variation of the

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selling price is explained by its size. That means

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the size of the house contributes about 90% to

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explain the variability of the selling price. So

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we would like to have R squared to be large

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enough. Now, R squared for simple regression only

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is given by this equation, correlation between X

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and Y squared.

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So, if we have the correlation between X and Y, and

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then you just square this value, that will give

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the correlation or the coefficient of

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determination. So simply, the determination

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coefficient is just the square of the correlation

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between X and Y. We know that R ranges between

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minus 1 and plus 1.

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So, R squared should be ranges between 0 and 1,

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because the minus sign will be cancelled, since we are

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squared is 90%, it means some, not all, the 

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variation Y is explained by the variation X. And 

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the remaining percent in this case, which is 10%, 

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this one due to, as I mentioned, maybe there 

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exists some other variables that affect the 

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selling price besides its size, maybe location of 

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the house affects its selling price. So R squared 

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is always between 0 and 1, it's always positive. R 

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squared equals 0, that only happens if there is no 

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linear relationship between Y and X. Since R is 0,

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then R squared equals 0. That means the value of Y 

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does not depend on X. Because here, as X 

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increases, Y stays nearly in the same position. It 

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means as X increases, Y stays the same, constant. 

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So that means there is no relationship or actually 

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there is no linear relationship because it could 

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be there exists non-linear relationship. But here 

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we are. Just focusing on linear relationship 

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between X and Y. So if R is zero, that means the 

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value of Y does not depend on the value of X. So

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as X increases, Y is constant. Now for the 

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previous example, R was 0.7621. To determine the

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coefficient of determination, One more time,

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square this value, that's only valid for simple 

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linear regression. Otherwise, you cannot square 

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the value of R in order to determine the 

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coefficient of determination. So again, this is 

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only true for 

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simple linear regression. 

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So R squared is 0.7621 squared will give 0.5808.

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Now, the meaning of this value, first you have to 

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multiply this by 100. So 58.08% of the variation 

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in house prices is explained by the variation in 

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square feet. So 58, around 0.08% of the variation

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in size of the house, I'm sorry, in the price is 

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explained by

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its size. So size by itself. Size only explains 

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around 50-80% of the selling price of a house. Now 

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the remaining percent which is around, this is the 

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error, or the remaining percent, this one is due

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to other variables, other independent variables. 

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That might affect the change of price.

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But since the size of the house explains 58%, that 

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means it's a significant variable. Now, if we add 

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more variables, to the regression equation for 

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sure this value will be increased. So maybe 60 or 

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65 or 67 and so on. But 60% or 50 is more enough 

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sometimes. But R squared, as R squared increases,

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it means we have good fit of the model. That means 

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the model is accurate to determine or to make some

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prediction. So that's for the coefficient of 

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determination. Any question? So we covered simple

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linear regression model. We know now how can we 

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compute the values of B0 and B1. We can state or 

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write the regression equation, and we can do some 

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interpretation about P0 and P1, making

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predictions, and make some comments about the

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coefficient of determination. That's all. So I'm 

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going to stop now, and I will give some time to

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discuss some practice.