Question

use a graphing calculator to create a scatter plot and find the line of best fit for the following data. round to the nearest hundredth.

the equation for the line of best fit is

Explanation:

Step1: Enter data into calculator

Enter the \(x\)-values (\(50,100,150,200,250,300,350\)) and \(y\)-values (\(50,90,160,190,240,330,320\)) into the graphing calculator's list function.

Step2: Calculate linear regression

Use the linear regression (\(LinReg(ax + b)\)) function on the calculator.
Let \(n = 7\).
The formula for the slope \(a=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\) and the \(y\)-intercept \(b=\overline{y}-a\overline{x}\), where \(\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\) and \(\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}\).
\(\sum_{i=1}^{7}x_i=50 + 100+150+200+250+300+350=1400\)
\(\sum_{i = 1}^{7}y_i=50 + 90+160+190+240+330+320 = 1380\)
\(\sum_{i=1}^{7}x_i^{2}=50^{2}+100^{2}+150^{2}+200^{2}+250^{2}+300^{2}+350^{2}=50^{2}(1 + 4+9+16+25+36+49)=2500\times140 = 350000\)
\(\sum_{i=1}^{7}x_iy_i=(50\times50)+(100\times90)+(150\times160)+(200\times190)+(250\times240)+(300\times330)+(350\times320)\)
\(=2500+9000 + 24000+38000+60000+99000+112000=344500\)
\(a=\frac{7\times344500-1400\times1380}{7\times350000 - 1400^{2}}\)
\(=\frac{2411500-1932000}{2450000 - 1960000}=\frac{479500}{490000}\approx0.98\)
\(\overline{x}=\frac{1400}{7}=200\), \(\overline{y}=\frac{1380}{7}\approx197.14\)
\(b = 197.14-0.98\times200=197.14 - 196=1.14\)

Answer:

The equation for the line of best fit is \(y = 0.98x+1.14\)