H2 Maths Notes (JC 1-2): 6.6) Correlation and Linear Regression

Before you revise\ Revisit scatter diagram basics and variance formulas so the transition to algebraic PMCC and regression is smooth. Keep a GC or spreadsheet handy to compute \( r \) and regression coefficients quickly.

Core Definitions

• Product-moment correlation coefficient \( r \) measures linear association between \( x \) and \( y \); \( -1 \leq r \leq 1 \).
• \( r > 0 \) indicates positive association, \( r < 0 \) indicates negative association, and \( |r| \approx 1 \) signals strong linearity.
• Least-squares regression line of \( y \) on \( x \) minimises \( \sum (y_i - \hat{y_i})^2 \) and has equation \( y - \bar{y} = b(x - \bar{x}) \), where \( b = r \frac{S_y}{S_x} \).
• \( S_x \) and \( S_y \) are sample standard deviations of \( x \) and \( y \) respectively.

Computing \( r \)

For paired data \( (x_i, y_i) \), \( r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{(n - 1) S_x S_y}).

Example -- Physics vs Maths scores

Data (Physics, Maths) for 6 students: (68, 72), (74, 75), (65, 69), (80, 83), (70, 73), (78, 81).

1. Enter into GC lists and run LinReg(ax+b).
2. Output: \( r = 0.987 \) (rounded), \( \bar{x} = 72.5 \), \( \bar{y} = 75.5 \).
3. Strong positive linear association.

Regression Line and Prediction

Using the same data, GC returns regression line \( y = 0.98x + 3.6 \) (values illustrative).

• To predict Maths score when Physics = 76: substitute \( x = 76 \), obtaining \( y = 0.98 \times 76 + 3.6 = 77.0 \) (nearest whole number).
• Only predict within the range of observed \( x \) (interpolation). Extrapolation is unreliable.

Residuals and Coefficient of Determination

• Residual: \( e_i = y_i - \hat{y_i} \); plot residuals to check linear model adequacy.
• Coefficient of determination \( r^2 \) gives the proportion of variance in \( y \) explained by \( x \).

Example -- Interpretation

If \( r = 0.82 \), then \( r^2 = 0.6724 \). State: “About 67% of the variation in Maths marks is explained by Physics marks via the fitted linear model.”

Hypothesis Test for \( r \)

• Test \( H_0: \rho = 0 \) vs \( H_1: \rho \neq 0 \) using t-statistic \( T = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}} \) with \( n - 2 \) degrees of freedom.
• Critical region depends on significance level; or compute p-value via calculator.

Example -- Significance of correlation

With \( n = 12 \) data points and \( r = 0.58 \), test at 5% whether \( \rho \neq 0 \).

1. \( T = \frac{0.58 \sqrt{10}}{\sqrt{1 - 0.58^2}} = 2.21 \).
2. Critical value \( t_{0.975,,10} = 2.228 \).
3. Since \( 2.21 < 2.228 \), fail to reject \( H_0 \); correlation not significant at 5%.

Calculator Workflows

• TI: LinReg(ax+b) returns \( a \) (gradient), \( b \) (intercept), \( r \), and \( r^2 \) when diagnostics are on.
• Casio: REG mode \(\rightarrow\) LR \(\rightarrow\) AX+B provides coefficients and \( r \).
• Use residual lists to draw scatter plots of \( x \) vs residuals; randomness indicates a good linear fit.
• Always state variables: “Let x = Physics score, y = Maths score.”

Exam Watch Points

• Label axes and highlight whether you are predicting \( y \) from \( x \) or vice versa. The regression line of \( x \) on \( y \) is different.
• Interpret \( r \) in words (“strong/weak, positive/negative”) and link back to context.
• Check units: regression line must retain units of \( y \) on the left-hand side.
• Do not claim causation; correlation only addresses association.
• Mention interpolation vs extrapolation explicitly when commenting on prediction reliability.

Quick Revision Checklist

• [ ] Compute \( r \) and regression coefficients quickly with calculator support.
• [ ] Write regression equations in the form \( y = ax + b \) and perform predictions.
• [ ] Interpret \( r^2 \) and residual plots qualitatively.
• [ ] Conduct and explain hypothesis tests for correlation with t-statistics.

Next steps: Practise past-year linear regression questions that combine with hypothesis tests or sampling intervals from Topics 6.4 and 6.5.