Linear Regression Lines (Cambridge International IGCSE Maths)

Linear Regression Lines

What is a linear regression line?

• Statistical software can calculate the equation for an 'ideal' line of best fit

  • This 'ideal' line of best fit is known as a regression line

    • It is more accurate than a line of best fit drawn by eye

  • You will be expected to use your graphic display calculator to generate the equation of linear regression

    • You do not need to calculate the equation by hand

• An equation of a linear regression line is usually given in the form $y=mx+b$

  • $b$ is the y-intercept of the regression line
  • $m$ is the gradient of the regression line
  • Both of those have the same meaning that they do for any line of best fit

How do I find the equation of a regression line using my graphic display calculator?

• You can find the equation of a line of regression using your graphic display calculator by doing the following:

• STEP 1 Add your raw data to a spreadsheet in your graphic display calculator

  • On some models, you must enter it in statistics mode
  • The data should be added as two columns
  • Label the columns if you are able (e.g. time, length) or x and y

• STEP 2 From the statistics function on your graphic display calculator, select the statistics calculations

  • Choose the option linear regressionequation``(m x+b)

  • On some models you instead select CALC followed by REG

    • You must also then select X, as we are finding a linear model, rather than quadratic or cubic etc
    • Then select ax+b for the format of the equation

• STEP 3 Assign the correct columns into your spreadsheet for the x and y lists

  • This is why labelling your columns at the start is important
  • Some models may simply assume the first column contains the x values

• STEP 4 Write down the values for equation$m$ and $b$produced by your calculator

  • Round each of the values to 3 significant figures (unless they can be written exactly)

• STEP 5 Substitute the values for $m$ and $b$ into the equation $y=mx+b$

How do I use and interpret the equation of a linear regression line?

• You may be asked to draw a regression line onto a scatter diagram

  • You need to know two points on the line

    • Choose two $x$ values (they don't need to correspond to any data values!)
    • Substitute these into the equation of the regression line to find the corresponding $y$ values

  • Plot these two points on the scatter diagram and draw a straight line through them

    • Use a ruler!

• A regression line drawn from its equation will always go through the mean point, $( \bar{x} , \bar{y} )$ for the data set

  • You may be required to use this fact in an exam question

Exam Hint

• Be careful with the $y=mx+b$ form of the regression line

  • It is the same as the $y=mx+c$ version of a straight line equation but uses the variable $b$ rather than $c$

Worked Example Rebecca, a regular jogger, recorded the number of calories she was able to burn ($y$ calories) by running different distances ($x$ km). This data is shown in the table and on the scatter diagram below.

Distance run (km)	10	2	7	6	12	14	5
Calories burned	620	180	438	366	830	870	equation315

(a) Find the equation of the regression line for $y$ in terms of $x$.

Enter the data as two columns in a spreadsheet or statistics mode on your graphic display calculator Label the columns if you are able to; x (distance run) and y (calories burned)

$x$	$y$
10	620
2	180
7	438
6	366
12	830
14	870
5	315

Select the stats calculation from the statistics function Choose the option for the linear regression (m x+b) Select the column that you input the distance data as the x list and the column that you input the calories burned data as the y list

On some models you instead select CALC followed by REG, then X for a linear model, and finally select ax+b for the format of the equation

Write down the values for $m$ and $b$ and round to 3 significant figures

$m=62.12264...\approx62.1\,b=20.01886...\approx20.0\,$

$y=62.1x+20.0$

(b) Draw the regression line on the scatter diagram.

Find the coordinates of two points on the line and draw the line through these points

$\text{when} x = 0 , y = 62.1 ( 0 ) + 20.0 = 20.0$

$> \text{when} x = 10 , y = 62.1 ( 10 ) + 20.0 = 641$

So draw the line through the points (0, 20) and (10, 641)

The mean of the data values for the distance run is 8 km.

(b) Use this information to find the mean of the data values for the calories burned.

Use the fact that the regression line always goes through the mean point $( \bar{x} , \bar{y} )$

Draw a vertical line up from 8 on the x- axis until it hits the regression line Then draw a horizontal line from there until it hits the y- axis

Read the value off the y- axis (it's a little bit less than 520)

Final answer: 516 calories

Marks would be awarded for a range of answers around that value