Question

use linear regression to find the equation for the linear function that best fits this data. round to two decimal places.
y =
basic funcs trig
x
hint
s
( ) | | π ∞ dne
← →
x 1 2 3 4 5 6
y 862 1093 1264 1587 1953 2550

Explanation:

Step1: Calculate sums

Let \(n = 6\). Calculate \(\sum_{i = 1}^{n}x_{i}=1 + 2+3 + 4+5 + 6=21\), \(\sum_{i = 1}^{n}y_{i}=862+1093 + 1264+1587+1953+2550 = 9319\), \(\sum_{i = 1}^{n}x_{i}^{2}=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}=1 + 4+9+16+25+36 = 91\), \(\sum_{i = 1}^{n}x_{i}y_{i}=1\times862+2\times1093+3\times1264+4\times1587+5\times1953+6\times2550=862 + 2186+3792+6348+9765+15300=38253\).

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the regression - line is \(m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}\).
Substitute the values: \(m=\frac{6\times38253-21\times9319}{6\times91 - 21^{2}}=\frac{229518-195699}{546 - 441}=\frac{33819}{105}\approx322.09\).

Step3: Calculate intercept \(b\)

The formula for the intercept \(b\) is \(b=\frac{\sum_{i = 1}^{n}y_{i}-m\sum_{i = 1}^{n}x_{i}}{n}\).
\(b=\frac{9319-322.09\times21}{6}=\frac{9319 - 6763.89}{6}=\frac{2555.11}{6}\approx425.85\).

Answer:

\(y = 322.09x+425.85\)