Question

Compute the appropriate regression equation for the following data : \[ \begin{array}{c|c|c|c|c|c} X & 1 & 2 & 3 & 4 & 5 \\ \hline Y & 5 & 7 & 9 & 11 & 13 \\ \end{array} \] X is the independent variable and Y is the dependent variable.

Hint:

The formula for \(b_{yx}\) is central to finding the regression line. A good way to check your calculation is to see if the line passes through the point of means, \((\bar{X}, \bar{Y})\). Here, if \(X=3\), \(Y = 3 + 2(3) = 9\), which matches our mean \( \bar{Y} \).

Answer 1

Since X is the independent variable and Y is the dependent variable, we need to find the regression equation of Y on X, which is given by \[ Y_c = a + b_{yx}X. \] We first prepare a table to calculate the necessary sums: \[ \begin{array}{|c|c|c|c|} \hline \mathbf{X} & \mathbf{Y} & \mathbf{X^2} & \mathbf{XY} \\ \hline 1 & 5 & 1 & 5 \\ 2 & 7 & 4 & 14 \\ 3 & 9 & 9 & 27 \\ 4 & 11 & 16 & 44 \\ 5 & 13 & 25 & 65 \\ \hline \sum X = 15 & \sum Y = 45 & \sum X^2 = 55 & \sum XY = 155 \\ \hline \end{array} \] Here, \(n = 5\). First, we find the means: \[ \bar{X} = \frac{\sum X}{n} = \frac{15}{5} = 3 \] \[ \bar{Y} = \frac{\sum Y}{n} = \frac{45}{5} = 9 \] Next, we calculate the regression coefficient \( b_{yx} \): \[ b_{yx} = \frac{n\sum XY - (\sum X)(\sum Y)}{n\sum X^2 - (\sum X)^2} = \frac{5(155) - (15)(45)}{5(55) - (15)^2} = \frac{775 - 675}{275 - 225} = \frac{100}{50} = 2 \] Now, we calculate the constant \(a\): \[ a = \bar{Y} - b_{yx}\bar{X} = 9 - (2)(3) = 9 - 6 = 3 \] Therefore, the appropriate regression equation is: \[ Y = 3 + 2X \]