Question

use the table below.
what is the line of best fit?

Explanation:

Step1: Calculate the mean of \(x\) and \(y\)

\(\bar{x}=\frac{2 + 5+10+15+20+30}{6}=\frac{82}{6}\approx13.67\)
\(\bar{y}=\frac{1+25+21+32+41+?}{6}\) (assuming the last \(y\) - value is calculated correctly based on the line of best - fit formula). But using the formula for the line of best - fit \(y = mx + b\), we can also use the formula \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\)

Let's assume we use the formula \(y=mx + b\). If we substitute \(x = 2,y = 1\); \(x = 5,y = 25\); \(x = 10,y = 21\); \(x = 15,y = 32\); \(x = 20,y = 41\) into \(y=mx + b\)

First, calculate \(\sum_{i=1}^{5}x_i=2 + 5+10+15+20=52\), \(\sum_{i = 1}^{5}y_i=1+25+21+32+41 = 120\), \(\sum_{i=1}^{5}x_i^{2}=4 + 25+100+225+400 = 754\), \(\sum_{i=1}^{5}x_iy_i=2\times1+5\times25+10\times21+15\times32+20\times41=2+125+210+480+820 = 1637\)

\(n = 5\)

\(m=\frac{5\times1637-52\times120}{5\times754 - 52^{2}}=\frac{8185-6240}{3770 - 2704}=\frac{1945}{1066}\approx1.82\)

\(b=\bar{y}-m\bar{x}\), \(\bar{x}=\frac{52}{5} = 10.4\), \(\bar{y}=\frac{120}{5}=24\)

\(b=24-1.82\times10.4=24 - 18.928 = 5.072\approx5.02\)

Answer:

\(f(x)=1.82x + 5.02\)