Linear Regression

If strong linear correlation exists on a scatter diagram then the data can be modelled by a linear model
- Drawing lines of best fit by eye is not the best method as it can be difficult to judge the best position for the line
The least squares regression line is the line of best fit that minimises the sum of the squares of the gap between the line and each data value
- This is usually called the regression line of y on x
- It can be calculated by looking at the vertical distances between the line and the data values
The regression line of y on x is written in the form $y = a x + b$
a is the gradient of the line
- It represents the change in y for each individual unit change in x
  - If a is positive this means y increases by a for a unit increase in x
  - If a is negative this means y decreases by |a| for a unit increase in x
b is the y – intercept
- It shows the value of y when x is zero
You are expected to use your GDC to find the equation of the regression line
- Enter the bivariate data and choose the model “ax + b”
- Remember the mean point $(\bar{x}, \bar{y})$ will lie on the regression line

The equation of the regression line can be used to decide what type of correlation there is if there is no scatter diagram
- If a is positive then the data set has positive correlation
- If a is negative then the data set has negative correlation
The equation of the regression line can also be used to predict the value of a dependent variable (y) from an independent variable (x)
- The equation should only be used to make predictions for y
  - Using a y on x line to predict x is not always reliable
- Making a prediction within the range of the given data is called interpolation
  - This is usually reliable
  - The stronger the correlation the more reliable the prediction
- Making a prediction outside of the range of the given data is called extrapolation
  - This is much less reliable
- The prediction will be more reliable if the number of data values in the original sample set is bigger

Once you calculate the values of a and b store then in your GDC
- This means you can use the full display values rather than the rounded values when using the linear regression equation to predict values
- This avoids rounding errors

Barry is a music teacher. For 7 students, he records the time they spend practising per week ( $x$ hours) and their score in a test ( $y$ %).

Time ( $x$ )	2	5	6	7	10	11	12
Score ( $y$ )	11	49	55	75	63	68	82

Write down the equation of the regression line of

y

x

, giving your answer in the form

y = a x + b

where

a

and

b

are constants to be found.

4-2-3-ib-ai-sl-linear-regression-a-we-solution

Give an interpretation of the value of

a

4-2-3-ib-ai-sl-linear-regression-b-we-solution

Another of Barry’s students practises for 15 hours a week, estimate their score. Comment on the validity of this prediction.

4-2-3-ib-ai-sl-linear-regression-c-we-solution

Linear Regression (DP IB Maths: AI HL)