Skip to content### 4.4 Linear correlation of bivariate data

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**Content-specific conceptual understandings**

**Linear correlation of bivariate data.****Pearson’s product-moment correlation coefficient, r.**

Technology should be used to calculate r. However, hand calculations of r may enhance understanding.

Critical values of r will be given where appropriate.

Students should be aware that Pearson’s product moment correlation coefficient (r) is only meaningful for linear relationships.

**Scatter diagrams; lines of best fit, by eye, passing through the mean point.**

Positive, zero, negative; strong, weak, no correlation.

Students should be able to make the distinction between correlation and causation and know that correlation does not imply causation.

**Equation of the regression line of y on x.****Use of the equation of the regression line for prediction purposes.****Interpret the meaning of the parameters, a and b, in a linear regression y=ax+b.**

Technology should be used to find the equation.

Students should be aware:

• of the dangers of extrapolation

• that they cannot always reliably make a prediction of x from a value of y, when using a yon x line.

**Exercises**

- The maximum temperature
*T*, in degrees Celsius, in a park on six randomly selected days is shown in the following table. The table also shows the number of visitors,*N*, to the park on

each of those six days.The relationship between the variables can be modelled by the regression equation*N*=*aT*+*b*.- Find the value of
*a*and of*b* - Write down the value of
*r* - Use the regression equation to estimate the number of visitors on a day when the maximum temperature is 15ºC.

- Find the value of
- The following table shows values of ln
*x*and ln*y*.The relationship between ln*x*and ln*y*can be modelled by the regression equation ln*y*=*a*ln*x*+*b*.- Find the value of
*a*and of*b* - Use the regression equation to estimate the value of
*y*when*x*= 57

The relationship between

*x*and*y*can be modelled using the formula*y*=*kx**n*, where*k*≠ 0,*n*≠ 0,*n*≠ 1.- By expressing ln
*y*in terms of ln*x*, find the value of*n*and of*k*.

- Find the value of
- The following table shows the hand lengths and the heights of five athletes on a sports team.The relationship between x and y can be modelled by the regression line with equation y = ax + b.
- Find the value of
*a*and of - Write down the correlation coefficient.
- Another athlete on this sports team has a hand length of 5cm. Use the regression equation to estimate the height of this athlete.

- Find the value of
- Adam is a beekeeper who collected data about monthly honey production in his bee hives.The relationship between the variables is modelled by the regression line with equation
*P*=*aN*+*b*- Write down the value of
*a*and of*b* - Use this regression line to estimate the monthly honey production from a hive that has 270 bees

- Write down the value of
- The following table shows the mean weight,
*y*kg, of children who are*x*years old.The relationship between the variables is modelled by the regression line with equation*y*=*ax*+*b*.- Find the value of
*a*and of*b* - Write down the correlation coefficient.
- Use your equation to estimate the mean weight of a child that is 1.95 years old.

- Find the value of