question 8\nuse linear regression to find the equation for the linear function that best fits this data…

Explanation:

Step1: Calculate sums

Let $n = 6$. $\sum_{i = 1}^{n}x_i=1 + 2+3 + 4+5 + 6=\frac{6\times(6 + 1)}{2}=21$ $\sum_{i = 1}^{n}y_i=801+832 + 839+836+876+899 = 5083$ $\sum_{i = 1}^{n}x_i^2=1^2+2^2+3^2+4^2+5^2+6^2=\frac{6\times(6 + 1)\times(2\times6 + 1)}{6}=91$ $\sum_{i = 1}^{n}x_iy_i=1\times801+2\times832+3\times839+4\times836+5\times876+6\times899$ $=801+1664+2517+3344+4380+5394 = 18090$

Step2: Calculate slope $m$

The formula for the slope $m$ of the regression - line is $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}$ Substitute the values: $n\sum_{i = 1}^{n}x_iy_i=6\times18090 = 108540$ $\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i=21\times5083 = 106743$ $n\sum_{i = 1}^{n}x_i^2=6\times91 = 546$ $(\sum_{i = 1}^{n}x_i)^2=21^2 = 441$ $m=\frac{108540-106743}{546 - 441}=\frac{1797}{105}\approx17.11$

Step3: Calculate intercept $b$

The formula for the intercept $b$ is $b=\overline{y}-m\overline{x}$, where $\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{21}{6}=3.5$ and $\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{5083}{6}\approx847.17$ $b = 847.17-17.11\times3.5$ $b = 847.17-59.89$ $b\approx787.28$

Answer:

$y = 17.11x+787.28$