linear regression\nuse linear regression to find the equation for the linear function that best fits this…

Explanation:

Step1: Calculate the means of x and y

$n = 6$ $\bar{x}=\frac{1 + 2+3+4+5+6}{6}=\frac{21}{6}=3.5$ $\bar{y}=\frac{105 + 124+136+157+179+204}{6}=\frac{905}{6}\approx150.83$

Step2: Calculate the numerator and denominator for slope m

$\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=(1 - 3.5)(105-150.83)+(2 - 3.5)(124 - 150.83)+(3 - 3.5)(136-150.83)+(4 - 3.5)(157-150.83)+(5 - 3.5)(179-150.83)+(6 - 3.5)(204-150.83)$ $=(- 2.5)(-45.83)+(-1.5)(-26.83)+(-0.5)(-14.83)+(0.5)(6.17)+(1.5)(28.17)+(2.5)(53.17)$ $=114.575 + 40.245+7.415 + 3.085+42.255+132.925$ $=340.5$ $\sum_{i = 1}^{n}(x_i-\bar{x})^2=(1 - 3.5)^2+(2 - 3.5)^2+(3 - 3.5)^2+(4 - 3.5)^2+(5 - 3.5)^2+(6 - 3.5)^2$ $=(-2.5)^2+(-1.5)^2+(-0.5)^2+(0.5)^2+(1.5)^2+(2.5)^2$ $=6.25+2.25 + 0.25+0.25+2.25+6.25$ $=17.5$ $m=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}=\frac{340.5}{17.5}\approx19.46$

Step3: Calculate the y - intercept b

$b=\bar{y}-m\bar{x}$ $b = 150.83-19.46\times3.5$ $b=150.83 - 68.11$ $b = 82.72$

Answer:

$y = 19.46x+82.72$