The table shows the numbers of new-vehicle sales (in...

Key Concepts

Correlation

Correlation measures the strength and direction of a linear relationship between two variables. In the context of linear regression, understanding the correlation between the variables helps in assessing how well the independent variable can explain the variability in the dependent variable, providing a basis for the reliability of the predictions made by the regression model.

Predictive Modeling

Predictive modeling involves using the regression equation to estimate the value of the dependent variable based on new or existing values of the independent variable. This process is fundamental in many fields, as it provides a quantitative method to forecast future outcomes and assess potential scenarios.

Linear Regression

Linear regression is a statistical technique used to model the relationship between a dependent variable and an independent variable by fitting a straight line to observed data. This method helps in understanding how changes in the independent variable affect the dependent variable and is widely used for prediction and forecasting.

Least Squares Method

The least squares method is the foundation for fitting the best possible regression line to a dataset. It works by minimizing the sum of the squared differences (residuals) between the observed values and the values predicted by the regression line, ensuring that the line accurately represents the overall trend in the data.

Regression Coefficients (Slope and Intercept)

Regression coefficients include the slope and the intercept of the regression line. The slope represents the average change in the dependent variable for each unit increase in the independent variable, while the intercept denotes the predicted value of the dependent variable when the independent variable is zero. Interpreting these coefficients is essential for understanding the underlying relationship between the variables.

Transcript

00:01 The following example studies the relationship between honda and toyota manufacturers or new car sales.

00:08 And we're asked to find the coefficient of determination and then also the standard error of the estimate.

00:15 So the way that you find the coefficient of determination is first you have to find the correlation coefficient, which is r.

00:21 And then you just square that and that gives you the coefficient of determination.

00:25 So we're going to do that using excel and the ti84.

00:28 I'm going to show you both ways.

00:29 If you do it manually, it really takes a long time.

00:31 You can do that, but i'm just to save time, i'm going to show you on the excel, and then i'll show you the shortcut on the ti -84.

00:37 And then the standard error, the estimate is where you use the regression lines.

00:40 Let's take a look at these formulas real quick.

00:42 So, n stands for the sample size.

00:44 In this case, i think there were 11 years worth of data values, so n equals 11, and then the summation of x times y.

00:51 So that's the honda times the toyota sales, and then minus the summation of x, so summation of, i think it's honda, and then times the summation of y, which i think is toyota.

01:01 It might be vice versa.

01:02 I can't remember.

01:04 And then, so that's the easy enough for the numerator.

01:06 And then the denominator is the square root of the sample size n times the summation of x squared.

01:12 So that's where we take each of those x values and square them, and then we add them all together.

01:16 And then minus the summation of x quantity squared.

01:19 So it's a little bit different.

01:20 We add up the xs together and then square the result.

01:23 So it's a very subtle difference there.

01:25 It's not the summation of x squared.

01:26 It's the summation of x quantity squared.

01:29 And then the same thing with the y.

01:31 So it's the sample size times the summation of all those toyota's squared.

01:35 So you square them and then add them.

01:36 But this one, we add them and then square them.

01:39 Okay, and then we're going to square all of that, and that'll give the cr squared.

01:42 The standard error of the estimate is the square root of the summation of the residual squared divided by the sample size minus 2.

01:50 So the residual is the actual minus the predicted.

01:52 And we use that regression line to predict the y value.

01:56 So why is the actual given.

01:57 Value in your table, but then y hat or y prime, just depends on what book you look at, but a y hat is the predicted.

02:06 So the residual is the actual minus the predicted, and then we square those, and then we add those together.

02:12 And then we divide that by n minus two, in this case we'll divide by nine, because that's the sample size is 11, and then we squared that result.

02:19 So let's get to it.

02:21 So here i have some formulas set up, so i went ahead and pre -populated the data.

02:25 So here's the x and here's the y.

02:26 Again, i think it's honda and then toyota.

02:30 It might be vice versa, but you just switch it.

02:34 The x, y column, so i made an xy column, and that's just simply the x times the y.

02:39 So i took the 1619 times the 1159, and i got that big number there.

02:43 So 8, so equals a2 times b2.

02:46 And you do that for all of them, but again, that's the nice thing about excel or technology.

02:51 You know, it does it for you.

02:52 Just double click and poof.

02:54 It's all that data.

02:55 Okay, so x squared is the same thing.

02:57 Thing.

02:57 So i do the 1619, this a2, and then i square the result and i get this big number.

03:02 And then i do that for all those x values, all those hondas.

03:05 And then y squared is the b2 squared.

03:09 So this is the toyota.

03:10 I square those.

03:11 And then double click and it gives me all those.

03:15 Okay, so that's the nuts and bolts of it.

03:17 That's really all i need.

03:18 Now i'm going to start doing some summation stuff.

03:21 So i do some x.

03:22 I call this sum x and that's the 22218.

03:25 And basically all that is just adding all of these.

03:28 X is together.

03:29 So summation of a2 through a12.

03:33 And then underneath that i went ahead and squared that.

03:35 So that's my summation of x quantity squared where i add them all together and then square the result.

03:40 And this is the big number that i get.

03:41 So it's a16 squared.

03:44 Likewise with the y.

03:45 So again, i think this is toyota.

03:47 So the summation of y, i just sum the b2 all the way up to the b12.

03:52 And then i square that result for the summation of y quantity squared.

03:58 And then all these other ones i just add sum of xy there it is c2 to c12 sum of x squared it's d2 to d12 sum of y squared is e2 to e12 and that's all the stuff that i need now i can use that formula so the numerator and i did this formula in pieces just because it helps me because it's really confusing otherwise i think but the numerator is the n which is 11 times the summation of x y which is the c16 minus the sum of x which is the a 16 times the sum of x which sum of y, which is the b -16.

04:31 So that's the numerator.

04:32 So let's just kind of tuck this away.

04:33 We'll use it later.

04:35 And then the denominator is really a lot more complicated.

04:37 So i went ahead and broke it up in a few pieces.

04:39 So i did the first part, which is the n times sum of x squared minus the sum of x quantity squared.

04:45 I have all these numbers...