Question

the following data were collected.

x: 22, 35, 50, 18, 21, 35, 46

y: 17, 40, 52, 10, 14, 32, 50

use technology to determine the slope of the line of best fit. round the final answer to the thousandths place.

0.854

-0.854

1.391

-1.391

Explanation:

Step1: Input data into technology

Input the \(x\) - values \(22,25,50,18,21,35,40\) and \(y\) - values \(17,40,52,10,14,32,50\) into a statistical software or calculator (such as TI - 84, Excel, or Google Sheets).

Step2: Calculate the regression line

Use the linear regression function. In Excel, for example, if the \(x\) - values are in column \(A\) (from \(A1\) to \(A7\)) and \(y\) - values are in column \(B\) (from \(B1\) to \(B7\)), the formula for the slope is \(=SLOPE(B1:B7,A1:A7)\).
When calculating, we first find the following sums:
Let \(n = 7\).
\(\sum x=22 + 25+50+18+21+35+40=211\)
\(\sum y=17 + 40+52+10+14+32+50=215\)
\(\sum xy=(22\times17)+(25\times40)+(50\times52)+(18\times10)+(21\times14)+(35\times32)+(40\times50)=374+1000 + 2600+180+294+1120+2000=7568\)
\(\sum x^{2}=22^{2}+25^{2}+50^{2}+18^{2}+21^{2}+35^{2}+40^{2}=484+625+2500+324+441+1225+1600=7203\)

The formula for the slope \(m\) of the regression line \(y = mx + b\) is \(m=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}\)

Substitute \(n = 7\), \(\sum x = 211\), \(\sum y=215\), \(\sum xy = 7568\), \(\sum x^{2}=7203\) into the formula:

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Answer:

\(1.290\)