question 6 points 3\nestimate the line of best fit for the following bivariate data.\n|x|y|\n|12|85|\n|16|65|…

Answer:

Let $n = 6$. First, calculate the following sums: $\sum_{i = 1}^{n}x_i=12 + 16+18+22+28+34 = 120$ $\sum_{i = 1}^{n}y_i=85 + 65+50+45+35+15 = 295$ $\sum_{i = 1}^{n}x_i^2=12^2+16^2+18^2+22^2+28^2+34^2=144 + 256+324+484+784+1156 = 3148$ $\sum_{i = 1}^{n}x_iy_i=12\times85+16\times65+18\times50+22\times45+28\times35+34\times15=1020+1040+900+990+980+510 = 5440$

The slope $m$ of the line of best - fit is given by the formula: [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}] [m=\frac{6\times5440 - 120\times295}{6\times3148-120^2}] [m=\frac{32640-35400}{18888 - 14400}] [m=\frac{- 2760}{4488}\approx - 0.615]

The mean of $x$ values $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{120}{6}=20$ The mean of $y$ values $\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{295}{6}\approx49.17$

The $y$-intercept $b$ is given by $b=\bar{y}-m\bar{x}$ [b = 49.17-(-0.615)\times20] [b = 49.17 + 12.3] [b=61.47]

The equation of the line of best - fit is $y=-0.615x + 61.47$, slope $m\approx - 0.615$ and $y$-intercept $b\approx61.47$

Explanation:

Step1: Calculate sums

Calculate $\sum_{i = 1}^{n}x_i$, $\sum_{i = 1}^{n}y_i$, $\sum_{i = 1}^{n}x_i^2$ and $\sum_{i = 1}^{n}x_iy_i$.

Step2: Find slope formula

Use the formula $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}$ to find the slope.

Step3: Calculate slope

Substitute the sum values into the slope formula.

Step4: Calculate means

Find $\bar{x}$ and $\bar{y}$.

Step5: Find y - intercept formula

Use the formula $b=\bar{y}-m\bar{x}$ to find the $y$-intercept.

Step6: Calculate y - intercept

Substitute $m$, $\bar{x}$ and $\bar{y}$ into the $y$-intercept formula.