Question

what is the equation of the least squares regression line?

variable x	2	7	25	13	4	15	33	10	19	27
variable y	16	19	22	21	19	19	26	21	22	25

○ $y = 16.9x+0.26$
○ $y = 0.92x + 16.9$
○ $y = 0.26x+16.9$
○ $y = 0.26x - 16.9$

Explanation:

Step1: Calculate means of x and y

Let $n = 10$.
$\bar{x}=\frac{2 + 7+25+13+4+15+33+10+19+27}{10}=\frac{155}{10} = 15.5$
$\bar{y}=\frac{16 + 19+22+21+19+19+26+21+22+25}{10}=\frac{200}{10}=20$

Step2: Calculate numerator and denominator for slope

$S_{xy}=\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})$
$S_{xx}=\sum_{i = 1}^{n}(x_i-\bar{x})^2$
$S_{xy}=(2 - 15.5)(16 - 20)+(7 - 15.5)(19 - 20)+(25 - 15.5)(22 - 20)+(13 - 15.5)(21 - 20)+(4 - 15.5)(19 - 20)+(15 - 15.5)(19 - 20)+(33 - 15.5)(26 - 20)+(10 - 15.5)(21 - 20)+(19 - 15.5)(22 - 20)+(27 - 15.5)(25 - 20)$
$S_{xy}=(- 13.5)\times(-4)+(-8.5)\times(-1)+9.5\times2+(-2.5)\times1+(-11.5)\times(-1)+(-0.5)\times(-1)+17.5\times6+(-5.5)\times1+3.5\times2+11.5\times5$
$S_{xy}=54 + 8.5+19-2.5 + 11.5+0.5+105-5.5+7+57.5$
$S_{xy}=265$
$S_{xx}=(2 - 15.5)^2+(7 - 15.5)^2+(25 - 15.5)^2+(13 - 15.5)^2+(4 - 15.5)^2+(15 - 15.5)^2+(33 - 15.5)^2+(10 - 15.5)^2+(19 - 15.5)^2+(27 - 15.5)^2$
$S_{xx}=(-13.5)^2+(-8.5)^2+9.5^2+(-2.5)^2+(-11.5)^2+(-0.5)^2+17.5^2+(-5.5)^2+3.5^2+11.5^2$
$S_{xx}=182.25 + 72.25+90.25+6.25+132.25+0.25+306.25+30.25+12.25+132.25$
$S_{xx}=964.5$
The slope $b=\frac{S_{xy}}{S_{xx}}=\frac{265}{964.5}\approx0.26$

Step3: Calculate y - intercept

The y - intercept $a=\bar{y}-b\bar{x}=20-0.26\times15.5=20 - 4.03=16.97\approx16.9$
The equation of the least - squares regression line is $y = 0.26x+16.9$

Answer:

$y = 0.26x + 16.9$ (corresponding to the third option)