Chapter 4.6, Problem 18HP

To determine

To find, how are lines of fit and linear regression similar or different.

Answer to Problem 18HP

They are similar in a sense that they represent a linear modal for the data. They are different because linear regression is more accurate as it utilizes a graphing calculator to find the equation of a line.

Explanation of Solution

Given information: Linear regression and lines of fit.

Formula used:

The equation for the regression line is given below.

  $\begin{array}{c}y=ax+b.\\ \text{Where},a=\frac{n{\displaystyle \sum xy-({\displaystyle \sum x)({\displaystyle \sum y)}}}}{n{\displaystyle \sum x^2-({\displaystyle \sum x{)}^2}}}.\\ b=\frac{{\displaystyle \sum y-a{\displaystyle \sum x}}}{n}.\\ \text{Where, }\text{n=number}\text{of}\text{observations}.\end{array}$

Calculation:

They are similar in a sense that they represent a linear modal for the data. They are different because linear regression is more accurate as it utilize a graphing calculator to find the equation of a line. However, you could have numerous lines of fit, while linear regression results in one line of best fit. If linear regression is used, can also use the

Correlation coefficient to see how closely the model fits the data. For example, the scatter plot below displays the points (1, 10), (2, 17), (3, 15), (4, 20), (5, 28), (6, 19), and (7, 25).

  

A line of fit could be drawn through (1,10) and (5,25).The equation of this line of fit is $y=(15/4)x+(25/4).$

Another line of fit may pass through (2, 15) and (7, 25). The equation of this line of fit is $y=2x+11$ .

More lines of fit that come close to the points could be drawn.

Use a calculator to find equation of the regression line for this data.

The only regression line for this data has an equation of about $y=2.2x+10.3.$