Question

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 ý = 0.94 +0.893. In the "Calculations" table are calculations involving the observed y values, the mean y of these values, and the values ♡ predicted from the regression equation. Sample data Calculations y 0.8 2.1 2.1 3.5 1.7956 1.7950 1.7 2.7 3.1969 0,9742 0.0094 0.6288 4.1575 به به هر 0.2007 0.1246 0.0246 0.5373 0.2714 1- 3.7 3.5 0.0036 0.0036 6.5530 5.1 6.0 Column sums 8.9669 1.1587 10.1520 Figure 1

Answer 1

Solution :

(a). The least square regression line given in the question is said to be a line which "best fits" tha sample data. The term "best fits" used because the line has an equation that minimizes the variation from least sqaure regression line which for these data is 1.1587.

(b). For the data point (0.8, 2.1), the value of residual is, y - y^.

y^ = 0.94 + 0.89*0.8 = 1.652. So the residual is, y - y^ = 2.1 - 1.652 = 0.448  $\approx$ 0.045. Residual at point (0.8, 2.1) is, 0.45.

(c). The total variation in sample y values is given by the sample data, which for this data is, 10.1520.

(d). The proportion of total variation in the sample y values that can be explained by the Linear relationship between the x and y is, R-squared = SSR / SST = Sum( y^- y.bar)^2 / Sum(y - y.bar)^2 = 8.9669 / 10.1520 = 0.88326  $\approx$ 0.88.

The proportion of total variation in the sample y values that can be explained by the Linear relationship between the x and y is, 0.88.