A full bivariate investigation practice questions | General Mathematics (Year 11, VCE Units 1 & 2, Bivariate data, Scatterplots)

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

b=r\times\frac{s_y}{s_x},\quad a=\bar{y}-b\bar{x}

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 x in hours to predict an assessment score y in points. Use the summary statistics and observed range to find the least-squares model, predict y and identify the type of prediction

r=0.75,\quad s_x=4,\quad s_y=12,\quad\bar{x}=18,\quad\bar{y}=60,\quad 10\le x\le24,\quad x=20

\(\displaystyle b=r\times\frac{s_y}{s_x}=0.75\times\frac{12}{4}=0.75\times3=2.25\)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

Swapping the regression roles

✕\(\displaystyle \widehat{\text{weekly revision time}}=19.5+2.25\times\text{assessment score}\)✓\(\displaystyle \widehat{\text{assessment score}}=19.5+2.25\times\text{weekly revision time}\)

The model predicts the response variable from the explanatory variable, so the roles must stay fixed

Rounding the slope before the intercept

✕\(\displaystyle b=2.25\to2.3,\quad a=60-2.3\times18=18.6\)✓\(\displaystyle a=60-2.25\times18=19.5\)

The unrounded slope must be used to calculate the intercept before any final rounding

Trusting an extrapolation

✕\(\displaystyle x=30\notin[10,24]\Rightarrow\text{interpolation and reliable}\)✓\(\displaystyle x=30\notin[10,24]\Rightarrow\text{extrapolation and not supported by the observed data}\)

A query outside the observed explanatory range is extrapolation and is not supported by the fitted data alone

Difficulty levels

standardComplete a rising-context investigation with the core correlation, line, interpretation and prediction chain

challengeComplete a falling-context investigation with an exterior query and additional model evidence

Questions about this topic

Why is the least-squares slope calculated before the intercept?+

The intercept uses the response mean minus the slope times the explanatory mean, so the slope is needed 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 11, VCE Units 1 & 2, General Mathematics, chapter 7: Bivariate data

In this chapter

Identify the explanatory and response variables Construct a scatterplot from a data table Read a scatterplot 7A Scatterplots: A full bivariate investigation Describe an association from a scatterplot Interpret an association in context 7B Describing association: A full bivariate investigation Properties and assumptions of Pearson's r Estimate r from a scatterplot Classify the strength of a linear association from r Association is not causation 7C Pearson's correlation coefficient: Compute r and the least-squares line from data 7C Pearson's correlation coefficient: A full bivariate investigation Find the equation of a line fitted to a scatterplot Residuals and what least squares minimises Compute the least squares line from summary statistics 7D Fitting linear models: Compute r and the least-squares line from data 7D Fitting linear models: A full bivariate investigation Interpret the intercept and slope of a fitted model Predict from a fitted model: interpolation and extrapolation 7E Model interpretation and prediction: A full bivariate investigation