# Explained and Unexplained variation and the least-squares regression line

Bivariate data obtained for the paired variables x and y are shown below, in the table labelled "Sample data." These data are plotted in the scatter plot in Figure 1, which also displays the least-squares regression line for the data. The equation for this line is y^ = 4.85+1.01x .

In the "Calculations" table are calculations involving the observed y values, the mean of these values, and

the values predicted from the regression equation.

X Y

225.2 240.3 1201.1770 63.9680 710.7556

245.2 232.8 209.0338 388.1688 1166.9056

255.8 279.7 14.0775 271.9861 162.3076

275.1 270.1 247.7791 271.9861 9.8595

297.6 311.9 1479.6332 158.7852 2019.6036

Column Sums 3151.7005 41.9127 4069.4320

1) For the data point (255.8, 279.7), the value of the residual is: (Round you answer to at least 2 decimal places.)

2) The least-squares regression line given above is said to be a line which "best fits" the sample data. The term "best fits" is used because the line has an equation that minimizes the _total sum of squares or regression sum of squares or error sum of squares (choose one), which for these data is __3151.7005, 924.827, 4069.4320. (choose one)

3) The value r2 is the proportion of the total variation is the sample y values that is explained by the estimated linear relationship between x and y. For these data, the value of r2 is:

4) The total variation is the sample y values is given by the _total sum of squares or regression sum of squares or error sum of squares, (choose one) which for these data is __3151.7005, 924.827, 4069.4320 (choose one).

#### Solution Summary

Step by step method for regression analysis is discussed here. Regression coefficients, coefficient of determination, scatter diagram and significance of regression model are explained in the solution.