Question

use technology to find an equation of the line of best fit for the data. round the slope and the y - intercept to the nearest hundredth.

x y
20 65
21 42
15 25
33 87
28 76
30 79

Explanation:

Step1: Enter data into technology

Input the \(x\) - values (\(20\), \(21\), \(15\), \(33\), \(28\), \(30\)) and \(y\) - values (\(65\), \(42\), \(25\), \(87\), \(76\), \(79\)) into a graphing calculator or statistical software.

Step2: Perform linear regression

Use the linear regression function. For most calculators, it's in the statistics menu. Let the calculator compute the slope (\(m\)) and \(y\) - intercept (\(b\)).
After running the linear regression on the data \((x_1,y_1)=(20,65)\), \((x_2,y_2)=(21,42)\), \((x_3,y_3)=(15,25)\), \((x_4,y_4)=(33,87)\), \((x_5,y_5)=(28,76)\), \((x_6,y_6)=(30,79)\):
The formula for the slope \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}\) and \(b=\overline{y}-m\overline{x}\), where \(n = 6\), \(\sum_{i=1}^{6}x_i=20 + 21+15+33+28+30=147\), \(\sum_{i=1}^{6}y_i=65 + 42+25+87+76+79=374\), \(\sum_{i=1}^{6}x_iy_i=(20\times65)+(21\times42)+(15\times25)+(33\times87)+(28\times76)+(30\times79)=1300+882 + 375+2871+2128+2370=9926\), \(\sum_{i=1}^{6}x_i^{2}=20^{2}+21^{2}+15^{2}+33^{2}+28^{2}+30^{2}=400+441+225+1089+784+900 = 3839\).
\(m=\frac{6\times9926-147\times374}{6\times3839 - 147^{2}}=\frac{59556-55078}{23034 - 21609}=\frac{4478}{1425}\approx3.14\)
\(\overline{x}=\frac{147}{6}=24.5\), \(\overline{y}=\frac{374}{6}\approx62.33\)
\(b = 62.33-3.14\times24.5=62.33 - 76.93=-14.60\)

Answer:

The equation of the line of best fit is \(y = 3.14x-14.60\)