The table shows the total square footages (in billions) of...

Key Concepts

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a straight line through the data. It is essential for prediction, estimation, and understanding the strength and direction of associations between variables.

Regression Equation

The regression equation, typically written in the form y = mx + b, represents the best-fitting line that minimizes the differences between observed values and the line's predictions. This equation is central to modeling the relationship between variables and is used to make inferences and predictions.

Dependent and Independent Variables

In regression analysis, the dependent variable is the outcome or response that one wishes to predict or explain, while the independent variable is the predictor or factor that is used to make predictions. Their relationship is analyzed to interpret how changes in the independent variable influence the dependent variable.

Interpretation of Slope

The slope in a regression equation quantifies the expected change in the dependent variable for each one-unit increase in the independent variable. It indicates the strength and direction (positive or negative) of the relationship between the two variables.

Interpretation of Intercept

The intercept in a regression equation represents the predicted value of the dependent variable when the independent variable is zero. It provides a baseline value in the model and is important for understanding the equation's complete behavior, even if the value of zero may not always be practically meaningful.

Transcript

00:01 The following is the solution to finding the r squared and the standard error of the estimate for a problem involving square footage and its correlation with sales price.

00:11 Okay, so to find the r squared, the formula is actually just r.

00:15 This is called the coefficient, the correlation coefficient.

00:18 And what we do to find the coefficient of determination r squared is you just square this thing.

00:22 So we're going to find, and i'm going to use excel for this, and then i'm going to show you a shortcut on the ti84 calculator.

00:26 But we use this formula using excel.

00:31 I'm going to tell you what this stuff means here in a second, and then we're going to square the result, and that will be our coefficient of determination.

00:37 And then the standard error, the estimates down here, a little easier to comprehend.

00:41 But here's the r.

00:42 So in the sensor sample size, this little sigma here, things that looks like an e, that means summation.

00:49 So it's a summation of x times y, so the pairs multiplied together, the ordered pairs multiplied together, minus the summation of the x's, times the summation of the y's, that's the numerator.

00:59 And then the denominator is the square root of n times the summation of x squared.

01:04 So you take each x and you square it, and you add those together.

01:07 And then this one a little bit different, it's the summation of x, and then you just square that.

01:11 So it's all the x is added together equals and then square the result.

01:16 And then that's times the same thing, but just for the y's.

01:18 So n times summation of y squared minus the summation of y quantity squared.

01:24 And then as for the standard error, the estimate is the square root of the.

01:26 Summation of y minus y prime or y hat and that's the actual minus the predicted squared so we add those together and then we divide by the sample size minus a two so i did this on excel and here are my x values this was i think square footage or something by maybe probably by thousands of feet or something and then this is the y i think it's sales price in billions or something so that's what was given to us and so the first thing i'm just looking at that r formula i need the x y so what i did is i just took a2 times b2 and that gave me this 4737 and what's nice about excel is you can just pop it in and that does it for all of those values.

02:07 Okay, so d part in this column d here x squared that's this whole column squared.

02:15 So that's why i did a two to the second power and i can do that a bunch of times and there it is and then like lies with the y squared.

02:22 I just take the the b column and i square each one of those and i do that for all 11 years.

02:30 Okay, so those are my main things.

02:32 So then down here, for sum of x, i just added, so i did, this is the function, equals sum a2 to a12, and that gives me these all added together.

02:42 And then sum of y, i took a, i'm sorry, b2 all the way up to b12, and i added those.

02:48 Some of x, y, i just added c2 to c12.

02:52 Sum of x squared is the sum of the d column.

02:57 And then sum of y squared.

02:58 Is the sum of the e column.

02:59 And then down here, i just took the sum of the x, which is the 65 .6, and then i squared that.

03:05 So that's why i did a16, which is this cell to the second power.

03:09 And then sum of y quantity squared is that.

03:12 So that's basically all the nuts and bolts of that massive formula for r.

03:17 And now i just made separate columns for the stuff.

03:21 Now for the formula, i kind of broke it up into pieces just to show you what's going on.

03:26 But this is my numerator here.

03:28 So that n times the sum of xy minus sum of x times sum of y.

03:32 And so to do that, that remember the sample size was 11.

03:35 And then i took c16, which is right here.

03:37 That's the sum of x, y, minus the sum of x, which was back here, the a16, times the sum of y, which was the b16.

03:46 So that was my numerator.

03:48 And then my denominator, i had to kind of go in pieces.

03:50 So i did the first part, that n times the x stuff.

03:54 So that's 11 times d16, which is right here.

04:00 So that's the sum of the x squared minus sum of x quantity squared.

04:04 So that's this thing right here.

04:05 That's why i did minus the a18.

04:08 And then likewise with the y did the same thing.

04:10 So it's the sample size times the y.

04:12 So e16 is the sum of y squared.

04:15 And then b18 is the sum of y quantity squared.

04:18 And then the denominator is just those two cells, multiply by each other.

04:24 So i 2 times j2, and then the actual denominator is the square root of that.

04:29 So i just did the square root of k2.

04:31 So i just kind of broke it up like that.

04:32 You don't have to do it that way, but i think it's easier to understand if you kind of go in pieces.

04:37 And then finally, to find the r, this is where we kind of put everything together.

04:40 We take the numerator, which is this cell here, that h2, divided by the denominator, or the square root of the denominator, which is this l2 here.

04:50 So that's why did h2 divide by l2? so that's my r, so 0 .9 .7.

04:53 So that's my r, so which is pretty significant.

04:55 And then r squared, i just squared that.

04:58 So my r squared is 99 .2%.

05:01 So what that means is i can explain 99 .2 % of the variation in y by x.

05:09 So in this case, i can explain 99 .2 % of the variation in sale price based on square footage...

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