Question

question 27
use linear regression to find the equation for the linear function that best fits this data. round to two decimal places.

x	1	2	3	4	5	6
y	563	583	621	648	643	663

y =
hint
check answer

Explanation:

Step1: Calculate sums

Let \(n = 6\). Calculate \(\sum_{i = 1}^{n}x_i=1 + 2+3 + 4+5 + 6=21\), \(\sum_{i = 1}^{n}y_i=563 + 583+621+648+643+663 = 3721\), \(\sum_{i = 1}^{n}x_i^2=1^2 + 2^2+3^2 + 4^2+5^2 + 6^2=1 + 4+9 + 16+25 + 36 = 91\), \(\sum_{i = 1}^{n}x_iy_i=1\times563+2\times583 + 3\times621+4\times648+5\times643+6\times663=563+1166+1863+2592+3215+3978 = 13377\).

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the regression line \(y=mx + b\) is \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}\).
Substitute the values: \(m=\frac{6\times13377-21\times3721}{6\times91 - 21^2}=\frac{80262-78141}{546 - 441}=\frac{2121}{105}\approx20.20\).

Step3: Calculate intercept \(b\)

The formula for the intercept \(b\) is \(b=\bar{y}-m\bar{x}\), where \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{21}{6}=3.5\) and \(\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{3721}{6}\approx620.17\).
\(b = 620.17-20.20\times3.5=620.17 - 70.7=549.47\).

Answer:

\(y = 20.20x+549.47\)