For the data in Exercise 6 of Section 10.2 "The Linear...

For the data in Exercise 6 of Section 10.2 "The Linear Correlation Coefficient" a. Compute the least squares regression line. b. Compute the sum of the squared errors $S S E$ using the definition $\Sigma\left(y-y^{\prime}\right)_{2}$. c. Compute the sum of the squared errors $S S E$ using the formula $S S E=S S_{y-\beta^{\prime}} \mid S S_{x y}$.

For the data in Exercise 6 of Section 10.2 "The Linear Correlation Coefficient"
a. Compute the least squares regression line.
b. Compute the sum of the squared errors $S S E$ using the definition $\Sigma\left(y-y^{\prime}\right)_{2}$.
c. Compute the sum of the squared errors $S S E$ using the formula $S S E=S S_{y-\beta^{\prime}} \mid S S_{x y}$.

Key Concepts

Least Squares Regression Line

The least squares regression line is a method in linear regression that fits a line through a set of data points by minimizing the sum of the squared vertical distances (errors) between the observed values and the values predicted by the line. This approach provides the best linear unbiased estimates for the intercept and slope, under the assumption that the errors are normally distributed with constant variance.

Sum of Squared Errors (SSE)

The Sum of Squared Errors (SSE) is a measure used in regression analysis to quantify the discrepancy between the observed data points and the values predicted by the regression model. It is calculated by summing the squares of the differences (residuals) between the observed dependent variable values and the values predicted by the regression line, thus providing a metric for the model's overall error.

Alternative Computation of SSE

An alternative method for computing SSE utilizes a formula that involves decomposing the total variability in the response variable. This method often includes components such as the total sum of squares and the regression sum of squares, linking the error term to the strength of the linear relationship between the independent and dependent variables. It also helps in understanding the proportion of variability explained by the model versus the unexplained variability.

Transcript

00:01 The following is a solution to number six, and we're asked to compute the least squares regression line from section 10 .2, number six, using that data.

00:07 So i'm going to use the ti84.

00:08 If you go to stat and then edit, you can see in l1, i put the x values 13558, and then in l2, i put the y values 5, negative 2, 2, negative 1, and negative 3.

00:19 So now if we go back to stat and then arrow over to calc, we're going to do the linear regression ax plus b.

00:25 So go to option 4, and your x list should be l1, and your y list.

00:30 This should be l2 and then we're going to calculate that and here's our regression line so y equals negative 0 .897x plus 4 .147 so let's go and write that down i should say y hat actually since it's a prediction so y hat is negative 0 .897 x plus 4 .147 okay so that's my least squares regression line then i'm supposed to find the sse using the summation of residual squared.

01:07 So it's y minus y hat, which is the actual y value minus the predicted y value squared.

01:13 That's called the residual squared.

01:15 And we're going to add those together, so the summation of that.

01:18 And i'm also going to use a calculator for this.

01:20 So if you go back to edit, we're going to use this third column.

01:24 We're going to apply the formula or the equation that we just found.

01:28 Or that we just found.

01:30 So in l3, so arrow over to l3, make sure l3 is highlighted, you can type in negative .897 and then times x, which in this case our x is going to l1.

01:44 So i'm going to say times l1 and then plus the 4 .147.

01:52 Whenever you do that, that should give you the y hats.

01:55 So these are the y hats.

01:57 These are the actual y.

01:58 So if i go back over to l4, an arrow up, so l4 is highlighted, now i need to find y minus y hat, which in this case would be l2 minus l3.

02:11 So second two minus second three, and that will give me my residuals.

02:20 Now i need to square them.

02:22 Now you could make that squared in l5 if you want, but we can actually just go to stat and then calc.

02:28 One bar stat and just change that to l4 and we can find the summation of of the squared's there.

02:34 So summation of x squared.

02:36 So this column is x.

02:38 Now it is.

02:39 Or we're calling it x at least.

02:40 So summation of x squared, that's the y minus y hat squared, is 20 .912...

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