If $\sum x _ { i } = 15 , \sum y _ { 2 } = 36 , \sum x _ { i } y _ { i } = 210 , \sum x _ { i } ^ { 2 } = 30 , \text { and } n = 20$ then the least squares estimate of the slope coefficient $\beta _ { 1 }$ of the true regression line $y = \beta _ { 0 } + \beta _ { 1 } x$ is
A) 18.75
B) 28.42
C) 9.15
D) 9.76
E) 10.50

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 $\sum x_i = 15$, $\sum y_i = 36$, $\sum x_iy_i = 210$, $\sum x_i^2 = 30$, and $n = 20$, we first calculate $\bar{x} = \frac{\sum x_i}{n} = \frac{15}{20} = 0.75$ and $\bar{y} = \frac{\sum y_i}{n} = \frac{36}{20} = 1.8$. Plugging these values into the formula gives $\beta_1 = \frac{210 - 20(0.75)(1.8)}{30 - 20(0.75)^2}$, which simplifies to $\beta_1 = 9.76$.

If ∑xi=15,∑y2=36,∑xiyi=210,∑xi2=30, and n=20\sum x _ { i } = 15 , \sum y _ { 2 } = 36 , \sum x _ { i } y _ { i } = 210 , \sum x _ { i } ^ { 2 } = 30 , \text { and } n = 20∑xi​=15,∑y2​=36,∑xi​yi​=210,∑xi2​=30, and n=20

If $\sum x _ { i } = 15 , \sum y _ { 2 } = 36 , \sum x _ { i } y _ { i } = 210 , \sum x _ { i } ^ { 2 } = 30 , \text { and } n = 20$