Least Squares Regression Calculator

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Example

Created on 2024-06-20Asked by Victoria Lee (Solvelet student)

Given the data points

(1, 2)

(2, 3)

(3, 5)

(4, 4)

, find the least squares regression line.

Solution

To find the least squares regression line

y = mx + b

, we need to calculate

m

and

b

using the given data points

(1, 2)

(2, 3)

(3, 5)

(4, 4)

. First, calculate the necessary sums:

\sum x_i = 1 + 2 + 3 + 4 = 10

\sum y_i = 2 + 3 + 5 + 4 = 14

\sum x_i y_i = (1)(2) + (2)(3) + (3)(5) + (4)(4) = 2 + 6 + 15 + 16 = 39

\sum x_i^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30

Next, calculate the slope

m

and intercept

b

m = \frac{n\sum x_i y_i - (\sum x_i)(\sum y_i)}{n\sum x_i^2 - (\sum x_i)^2}

m = \frac{4 \cdot 39 - 10 \cdot 14}{4 \cdot 30 - 10^2} = \frac{156 - 140}{120 - 100} = \frac{16}{20} = 0.8

b = \frac{\sum y_i - m \sum x_i}{n}

b = \frac{14 - 0.8 \cdot 10}{4} = \frac{14 - 8}{4} = \frac{6}{4} = 1.5

Therefore, the least squares regression line is:

y = 0.8x + 1.5

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Mateo Rodriguez on Solvelet

1. Perform simple linear regression on the dataset

\{(1, 2), (2, 3), (3, 4), (4, 5)\}

to predict the dependent variable

y

given the independent variable

x

,2. Determine the roots of the Legendre polynomial of degree

4

P_4(x)

DefinitionA linear regression line is a ‘line of best fit’ for the data in the sense that it minimizes the sum of the squares of the vertical distances of the points from the line. Least squares regression is a statistical technique that is used to determine the “best” linear relationship between one dependent variable and one or more independent variables by minimizing the sum of the squares of the differences between the observed and predicted values. For example for a dataset {xi,yi} then the least squares regression line y=mx+b will minimize the sum of squares ∑(yi−mxi+b)2 and it provides the best linear fit.

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