ETF1100 Chap.8 Correlation and Simple Regression
Correlation and Simple Regression
Correlation and Simple Regression begins with a scatterplot, then quantifies linear association and fits a prediction line. The chapter interprets slope and intercept with units, computes residuals as observed minus predicted, reads R-squared without turning it into accuracy, checks residual patterns, limits extrapolation and rejects causal claims unsupported by an observational design.
What this chapter covers
- 01
Scatterplot direction and form
- 02
Correlation strength and outliers
- 03
Slope and intercept interpretation
- 04
Predictions and residuals
- 05
R-squared and model fit
- 06
Extrapolation and causal limits
Worked example · free
Interpret a delivery regression
- 1Compute 18.4 + 0.72 x 50 = 54.4 minutes.
- 1Compute residual = 59 - 54.4 = +4.6 minutes.
- 1Interpret the slope as 0.72 predicted minute per additional kilometre.
- 1State that the observed trip was slower than predicted.
- 1Limit the association to the observed range and avoid causation.
Key terms
- Scatterplot
- A graph displaying paired numerical observations to reveal direction, form and unusual points.
- Correlation
- A unit-free measure of linear association ranging from minus one to one.
- Regression Slope
- The predicted outcome change for a one-unit predictor increase.
- Regression Intercept
- The predicted outcome when the predictor equals zero.
- Residual
- The observed outcome minus its fitted outcome.
- R-squared
- The proportion of observed outcome variation explained by the fitted model in the sample.
Correlation and Simple Regression FAQ
What should I describe in a scatterplot?
Describe direction, form, strength and unusual observations before quoting correlation. A relationship can be positive or negative, linear or curved, tight or weak, and influenced by outliers. Correlation alone can miss a strong nonlinear pattern.
How do I interpret a regression slope?
Name the predicted outcome, the one-unit increase in the predictor, the direction and both units. Restrict the sentence to the observed range and design. In multiple regression, add that the other included predictors are held constant.
Why does regression not prove causation?
An omitted variable can affect both predictor and outcome, reverse influence may be possible and the sample can be selected. A fitted line summarises association and prediction unless an appropriate design isolates a causal effect.
How should an influential observation be handled?
Verify the record and investigate its business origin before changing the dataset. Compare fitted results with and without it as a sensitivity analysis, but report that comparison transparently. A valid unusual case may reveal a separate process or model weakness; automatic deletion can conceal information and bias the fitted relationship.
Exam move
Start every relationship problem with a sketch or scatterplot description: direction, form, strength and unusual points. Learn correlation as a measure of linear association rather than a universal relationship score. For simple regression, write the fitted equation with units, then practise one prediction and one residual using observed minus predicted.
Build complete slope sentences that name the outcome, predictor, one-unit increase and predicted change. Treat the intercept cautiously when zero is outside the observed range. Explain R-squared as sample variation explained, never accuracy or causal share. Inspect residuals for curves, changing spread and influential cases. Mark predictions beyond the observed predictor range as extrapolations.
End each model answer with one omitted factor or design limitation. In automatic questions, separate the coefficient, standard error, test statistic and p-value rows before selecting a number. Recompute with full precision and round only the requested final entry.
Take one scatterplot and narrate it before fitting: identify which variable predicts, which responds, the visible range, direction, form, strength, clusters and unusual points. Predict the sign of correlation and slope, then compare with the output. Rewrite the slope using a meaningful predictor increment if one unit is too small for business use, while scaling the predicted change consistently.
Substitute two in-range values into the fitted line and calculate signed residuals so above-line and below-line cases become intuitive. Plot residuals against fitted values and look for curvature, a funnel or an isolated influential case; each pattern suggests a different model concern.
Refit without an unusual observation only as a sensitivity check, document the change and investigate the case rather than erasing it automatically. Mark the observed predictor range and label any outside prediction as extrapolation. Generate one example of a strong curved relationship with correlation near zero to break the habit of equating zero correlation with no relationship.
For every causal claim, list a plausible confounder and ask what design would identify an intervention effect. Evaluate prediction with errors on new cases rather than R-squared alone. Close with an equation, coefficient interpretation, prediction scope, fit statement, diagnostic finding and limitation. This sequence makes the regression output a model to critique rather than a table to quote.
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