Problem Statement
From the following bivariate data, calculate the two regression lines and estimate the value of $y$ when $x = 10$:
$$\sum x = 30, \quad \sum y = 40, \quad \sum x^2 = 220, \quad \sum y^2 = 340, \quad \sum xy = 214, \quad n = 5$$
Verified Solution & Marking Scheme
Calculate means x̄ and ȳ
$$\bar{x} = \frac{\sum x}{n} = \frac{30}{5} = 6$$
$$\bar{y} = \frac{\sum y}{n} = \frac{40}{5} = 8$$
Calculate regression coefficient b_yx
$$b_{yx} = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2} = \frac{5(214) - (30)(40)}{5(220) - (30)^2} = \frac{1070 - 1200}{1100 - 900} = \frac{-130}{200} = -0.65$$
Form regression line of y on x
$$y - \bar{y} = b_{yx}(x - \bar{x})$$
$$y - 8 = -0.65(x - 6) = -0.65x + 3.9$$
$$y = -0.65x + 11.9$$
Estimate y when x = 10
$$y = -0.65(10) + 11.9 = -6.5 + 11.9 = 5.4$$