If $\sum x _ { i } = 28 , \sum y _ { i } = 54 , \sum x _ { i } y _ { 2 } = 156 , \sum x _ { i } ^ { 2 } = 82 , \text { and } n = 10$ then the least squares estimate of the slope coefficient $\beta _ { 1 }$ of the true regression line $y = \beta _ { 0 } + \beta _ { 1 } x$ is
A) 3.60
B) 0.75
C) 1.33
D) 4.80
E) 1.68

Question

Quizplus · Accepted Answer

The least squares estimate of the slope coefficient $\beta_1$ in a simple linear regression is given by the formula $\beta_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$, which can also be expressed as $\beta_1 = \frac{\sum x_iy_i - n\bar{x}\bar{y}}{\sum x_i^2 - n\bar{x}^2}$. Given $\sum x_i = 28$, $\sum y_i = 54$, $\sum x_iy_i = 156$, $\sum x_i^2 = 82$, and $n = 10$, we first calculate $\bar{x} = \frac{\sum x_i}{n} = \frac{28}{10} = 2.8$ and $\bar{y} = \frac{\sum y_i}{n} = \frac{54}{10} = 5.4$. Then, $\beta_1 = \frac{156 - 10(2.8)(5.4)}{82 - 10(2.8)^2} = \frac{156 - 151.2}{82 - 78.4} = \frac{4.8}{3.6} = 1.33$.

If ∑xi=28,∑yi=54,∑xiy2=156,∑xi2=82, and n=10\sum x _ { i } = 28 , \sum y _ { i } = 54 , \sum x _ { i } y _ { 2 } = 156 , \sum x _ { i } ^ { 2 } = 82 , \text { and } n = 10∑xi​=28,∑yi​=54,∑xi​y2​=156,∑xi2​=82, and n=10

If $\sum x _ { i } = 28 , \sum y _ { i } = 54 , \sum x _ { i } y _ { 2 } = 156 , \sum x _ { i } ^ { 2 } = 82 , \text { and } n = 10$