Question

use linear regression to find the equation for the linear function that best fits this data. round both numbers to two decimal places. write your final answer in a form of an equation y = mx + b
linear regression equation:
linear regression

Explanation:

Step1: Calculate necessary sums

First, we list the data points: \((1, 69)\), \((2, 87)\), \((3, 105)\), \((4, 130)\), \((5, 134)\), \((6, 163)\)

Number of data points \(n = 6\)

Calculate \(\sum x\), \(\sum y\), \(\sum xy\), \(\sum x^2\)

\(\sum x = 1 + 2 + 3 + 4 + 5 + 6 = 21\)

\(\sum y = 69 + 87 + 105 + 130 + 134 + 163 = 688\)

\(\sum xy = (1\times69) + (2\times87) + (3\times105) + (4\times130) + (5\times134) + (6\times163)\)
\(= 69 + 174 + 315 + 520 + 670 + 978 = 2726\)

\(\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91\)

Step2: Calculate slope \(m\)

The formula for slope \(m\) in linear regression is:

\(m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}\)

Substitute the values:

\(n = 6\), \(\sum xy = 2726\), \(\sum x = 21\), \(\sum y = 688\), \(\sum x^2 = 91\)

\(m = \frac{6\times2726 - 21\times688}{6\times91 - 21^2}\)

First, calculate numerator: \(6\times2726 = 16356\), \(21\times688 = 14448\), so numerator \(= 16356 - 14448 = 1908\)

Denominator: \(6\times91 = 546\), \(21^2 = 441\), so denominator \(= 546 - 441 = 105\)

\(m = \frac{1908}{105} \approx 18.17\) (rounded to two decimal places)

Step3: Calculate y-intercept \(b\)

The formula for \(b\) is:

\(b = \frac{\sum y - m\sum x}{n}\)

Substitute the values:

\(b = \frac{688 - 18.17\times21}{6}\)

First, calculate \(18.17\times21 = 381.57\)

Then, \(688 - 381.57 = 306.43\)

\(b = \frac{306.43}{6} \approx 51.07\) (rounded to two decimal places)

Step4: Form the equation

The linear regression equation is \(y = mx + b\), substituting \(m \approx 18.17\) and \(b \approx 51.07\)

Answer:

\(y = 18.17x + 51.07\)