H2 Maths 6.6 - Correlation and Linear Regression

Step 1 - Set up the problem

Six students each sat a Physics test and a Maths test.

Step 1 - Set up the problem

We want to know: does a higher Physics mark tend to predict a higher Maths mark?

Step 1 - Set up the problem

Let x be the Physics score and y be the Maths score.

Step 2 - Compute the PMCC

Enter the data into your graphing calculator and run LinReg(ax+b).

Step 2 - Compute the PMCC

The output gives r equals 0.984, the product-moment correlation coefficient.

Step 2 - Compute the PMCC

Because r is close to 1, there is a strong positive linear association between Physics and Maths scores.

Step 3 - Find the regression line

The regression line of y on x minimises the sum of squared residuals.

Step 3 - Find the regression line

Its gradient is b equals r times S-y over S-x, and the line always passes through the mean point x-bar, y-bar.

Step 3 - Find the regression line

From the calculator output, the equation is y equals zero point nine one three x plus nine point three four.

Step 4 - Predict and check validity

Predict the Maths score for a student who scored 76 in Physics.

Step 4 - Predict and check validity

Substitute x equals 76 into the regression line.

Step 4 - Predict and check validity

The result is approximately 78.7, so we report 79 to the nearest whole number.

Step 4 - Predict and check validity

Since 76 lies inside the observed range of 65 to 80, this is interpolation and the prediction is reliable.

Step 5 - Interpret and state limitations

The coefficient of determination r-squared equals 0.968.

Step 5 - Interpret and state limitations

This means approximately 96.8 percent of the variation in Maths marks is explained by Physics marks via the fitted linear model.

Step 5 - Interpret and state limitations

Two key limitations: first, do not extrapolate beyond the data range.

Step 5 - Interpret and state limitations

Second, correlation does not imply causation - a high Physics mark may not cause a high Maths mark.