Back to Statistics and Probability

Higher Applications of Mathematics

Linear regression

Fitting and interpreting least-squares lines.

Statistics referenceStatistics mixed quiz

Statistics reference

Check probability, diagrams, averages, regression, intervals and tests.

Mixed revision quiz

Review Statistics and Probability with adaptive feedback.

Before you start

  • Be confident reading values from tables and graphs.
  • Check units, sample size and what each variable represents.
  • Use context in written answers, especially when interpreting results.

Method chooser

Which statistics method do I use?

Are there stages or repeated chance events?

Use a tree diagram.

Do categories overlap?

Use a Venn diagram.

Do you need a typical value?

Calculate an average.

Do you need to compare consistency?

Compare spread using standard deviation or IQR.

Are you comparing distributions?

Use box plots or histograms.

Are you checking a relationship?

Use scatter graph and correlation.

Do you need a prediction line?

Use linear regression.

Are you estimating a population value?

Use a confidence interval.

Are you judging evidence against a claim?

Use a hypothesis test.

Are the same people measured twice?

Use a paired t-test.

Are you comparing two proportions?

Use a z-test for two proportions.

Statistics lesson

Key idea

  • This topic focuses on using a fitted line to model and predict a numerical relationship. In Higher Applications, the aim is to use statistical methods to make careful decisions from real data.
  • Good statistical work has three parts: choose a suitable method, carry it out accurately, then explain what the result means in the situation.
  • When writing conclusions, use cautious language such as 'this suggests' or 'there is evidence to suggest'. Data can support a conclusion, but it rarely proves it completely.

Key formulae, definitions and methods

  • Regression line: y = a + bx, where b is the gradient and a is the intercept.
  • Gradient means the predicted change in y for one unit increase in x.
  • Use regression predictions only where the model is sensible.

Technology output practice

Interpreting statistical output

Read the simulated output, pick out the key value, then turn it into a written conclusion. This is a learning preview, not a real RStudio environment.

Context

Linear model output

A youth group models total ticket income from the number of tickets sold.

Simulated output

> model <- lm(income ~ tickets, data = sales)
> coefficients(model)
(Intercept)     tickets
     42.10        5.70

Fitted model: income = 42.10 + 5.70 x tickets

Intercept

42.10

The model's starting value when tickets is 0. It may not have a useful real-life meaning.

Slope

5.70

For each extra ticket, predicted income increases by about 5.70 pounds.

Prediction

Use with care

Predictions are more reliable inside the range of data used to fit the model.

What it means

The slope is the most useful value for explaining how the variables are linked. The intercept should be interpreted only if zero tickets makes sense in context.

What to write

The model predicts that each extra ticket sold increases income by about 5.70 pounds. Predictions should be treated cautiously if the number of tickets is outside the original data range because that would be extrapolation.

Weak answer: The answer is 42.10 because it is first in the table.

Watch out

Check which coefficient matches the question. For rate of change, use the slope beside the explanatory variable.

Which value tells you the increase in income per extra ticket?

Choose an option, then check the feedback.

Worked examples

Worked example 1

Choose the method

A college estimates printing cost from the number of pages in project booklets.

  1. Identify the explanatory variable x and response variable y.
  2. Use the regression equation supplied or generated by technology.
  3. Substitute the x-value carefully.

Regression turns a visible relationship into a practical prediction rule.

Worked example 2

Carry out and interpret

A college estimates printing cost from the number of pages in project booklets.

  1. Interpret the gradient in context.
  2. Interpret the intercept only if it makes practical sense.
  3. State the prediction with units.

The gradient is often the most useful part of the equation to interpret.

Worked example 3

Check the conclusion

A college estimates printing cost from the number of pages in project booklets.

  1. Check whether the prediction is interpolation or extrapolation.
  2. Consider scatter, outliers and context.
  3. Explain whether the model seems reliable.

A regression answer should include a reliability comment, not just a calculation.

Watch out

  • Choosing a method because it is familiar rather than because it matches the data.
  • Giving a numerical answer without explaining what it means in context.
  • Mixing up sample evidence with certainty about the whole population.
  • Ignoring outliers, skewness, units or the scale on a graph.
  • Using causal language when the data only shows association.

Technology connection

Related RStudio and Spreadsheet topics

RStudio: Linear regression in RStudio

Next step

Move into practice

Use the learning notes to choose suitable summaries and conclusions, then try varied data sets, tables, p-values and interpretation prompts.

Statistics mixed quiz