Mathematics Standard • Year 12 • Module 5 • Lesson 3
Pearson's Correlation Coefficient, Apply
Contextual Problem Set for Pearson's Correlation Coefficient. Work through each self-contained problem, show your reasoning, then open the worked answers to check your method.
1. Core skill
For five plants, hours of light x and height y (cm) are (1,12), (2,15), (3,17), (4,21), (5,23). A calculator reports r = 0.994. Interpret r in context and state what its value says about linearity. 3 marks
2. Use the idea
For another sample, r = −0.73 between outside temperature and hot-drink sales. Write a complete interpretation and one caution about generalising. 3 marks
3. Check your reasoning
An extreme outlier is added to a dataset and r changes from 0.78 to 0.24. Explain why this is possible and name a graphical check that should accompany r. 2 marks
Self-check
For each problem, check that your answer includes the requested variable, units or contextual interpretation.
1. Core skill (3 marks)
There is a very strong positive linear association between hours of light and plant height in these five observations. The value near +1 means the points lie close to an increasing straight-line pattern; it does not say the relationship is perfectly causal.
2. Use the idea (3 marks)
The sample shows a moderately strong to strong negative linear association: higher outside temperatures tend to accompany lower hot-drink sales. This does not prove temperature is the only cause, and the result may not generalise beyond the sampled locations or period.
3. Check your reasoning (2 marks)
Pearson’s r is sensitive to outliers because one extreme point can strongly affect the covariance and standard deviations used in r. Inspect a scatterplot for outliers, curvature and clusters before relying on the coefficient.