Run a bivariate investigation
A full investigation keeps one explanatory-to-response direction through the data description, correlation, fitted line, interpretation and prediction. The least-squares calculation below uses exact summary values before the final rounding
- Fix the variable roles Use the explanatory variable to predict the response variable throughout the investigation \(\displaystyle x=\text{EV},\quad y=\text{RV}\)
- Keep the paired data together The same explanatory and response values must stay paired when describing the association and fitting the model \(\displaystyle (x_i,y_i)=(\text{EV}_i,\text{RV}_i)\)
- Compute the slope first For a least-squares line, calculate the slope before the intercept because the intercept formula uses the slope \(\displaystyle b=r\times\frac{s_y}{s_x},\quad a=\bar{y}-b\bar{x}\)
- Check the observed range A prediction is interpolation when its explanatory value is within the observed range and extrapolation when it is outside \(\displaystyle x\in[x_{\min},x_{\max}]\Rightarrow\text{interpolation}\)
Worked example
A study uses weekly revision time in hours to predict an assessment score in points. The summary statistics are shown below. Observed weekly revision times range from 10 to 24 hours. Find the least-squares model and predict the score for 20 hours of revision, identifying the type of prediction
- \(\displaystyle b=r\times\frac{s_y}{s_x}=0.75\times\frac{12}{4}=0.75\times3=2.25\)Calculate the slope first because the intercept calculation needs b.
- \(\displaystyle a=\bar{y}-b\bar{x}=60-2.25\times18=60-40.5=19.5\)Use the unrounded slope to find the intercept
- \(\displaystyle \hat{y}=19.5+2.25x\)The fitted model predicts assessment score from weekly revision time
- \(\displaystyle \hat{y}(20)=19.5+2.25\times20=64.5,\quad10\le20\le24\Rightarrow\text{interpolation}\)The query value 20 lies inside the observed explanatory range, so this is interpolation
Common mistakes
Difficulty levels
Questions about this topic
Why is the least-squares slope calculated before the intercept?+
The intercept calculation uses the slope, so calculate the slope first
Why must the variable roles remain fixed?+
A fitted model predicts the response variable from the explanatory variable. Swapping the roles gives a different regression model
What makes a prediction interpolation?+
It is interpolation when the explanatory value used in the prediction lies within the observed explanatory-variable range
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, General Mathematics, chapter 3: Linear regression