Question

based on the data shown below, calculate the regression line (each value to at least two decimal places). x: 4, 5, 6, 7, 8, 9; y: 22.6, 20.48, 19.36, 16.84, 17.72, 15.8. question help: d post to forum

Explanation:

Step1: Calculate means of x and y

Let $x_i$ be the values of the $x -$ variable and $y_i$ be the values of the $y -$ variable.
$n = 6$
$\bar{x}=\frac{4 + 5+6+7+8+9}{6}=\frac{39}{6}=6.5$
$\bar{y}=\frac{22.6+20.48+19.36+16.84+17.72+15.8}{6}=\frac{112.8}{6}=18.8$

Step2: Calculate numerator and denominator for slope

The formula for the slope $b$ of the regression line $y = a+bx$ is $b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$
$\sum_{i = 1}^{6}(x_i-\bar{x})(y_i - \bar{y})=(4 - 6.5)(22.6-18.8)+(5 - 6.5)(20.48 - 18.8)+(6 - 6.5)(19.36 - 18.8)+(7 - 6.5)(16.84 - 18.8)+(8 - 6.5)(17.72 - 18.8)+(9 - 6.5)(15.8 - 18.8)$
$=(- 2.5)\times3.8+( - 1.5)\times1.68+( - 0.5)\times0.56+0.5\times(-1.96)+1.5\times(-1.08)+2.5\times(-3)$
$=-9.5-2.52 - 0.28-0.98-1.62 - 7.5=-22.4$
$\sum_{i=1}^{6}(x_i-\bar{x})^2=(4 - 6.5)^2+(5 - 6.5)^2+(6 - 6.5)^2+(7 - 6.5)^2+(8 - 6.5)^2+(9 - 6.5)^2$
$=(-2.5)^2+( - 1.5)^2+( - 0.5)^2+0.5^2+1.5^2+2.5^2$
$=6.25 + 2.25+0.25+0.25+2.25+6.25 = 17.5$
$b=\frac{-22.4}{17.5}=-1.28$

Step3: Calculate intercept

The formula for the intercept $a$ is $a=\bar{y}-b\bar{x}$
$a = 18.8-(-1.28)\times6.5=18.8 + 8.32=27.12$

Answer:

$y=27.12-1.28x$