Bivariate Statistics

**Hours studied and exam mark.** A teacher records the number of hours that 10 students revised, and their exam marks (out of 100): | Hours | 2 | 3 | 5 | 6 | 7 | 8 | 9 | 10 | 12 | 15 | |-------|---|---|---|---|---|---|---|----|----|----| | Mark | 35 | 40 | 50 | 55 | 60 | 65 | 70 | 70 | 80 | 90 | (a) Sketch a scatter plot. Describe the correlation. (b) Find the means $\bar{x}$ and $\bar{y}$. (c) Estimate the gradient of the line of best fit and write its equation, passing through the mean. (d) Predict the score for a student who revises 4 hours. Is this reliable?

**Spurious correlation.** Between 2000 and 2019, the number of films Nicolas Cage starred in correlates strongly with the number of swimming pool drownings in the USA ($r \approx 0.67$). (a) Does this mean Nicolas Cage films cause drownings? (b) Suggest two explanations for the correlation. (c) What does this example teach us about interpreting correlation?

**Predicting temperature.** A dataset of altitude (m) and temperature (°C) at noon yields the regression line $$T = -0.0065 a + 15.$$ (a) Interpret the gradient and intercept. (b) Predict $T$ at altitude 1500 m. (c) Predict $T$ at altitude 9000 m (Everest). Is this reliable?

**Identifying the outlier.** Eight data points $(x, y)$: $$(1, 3), (2, 5), (3, 7), (4, 8), (5, 10), (6, 12), (7, 25), (8, 16)$$ (a) Identify the outlier. (b) Compute the LOBF gradient with and without the outlier (informally). (c) Comment on the effect of the outlier.

**Causation vs correlation.** For each pair, suggest whether the correlation likely reflects causation, confounding, or coincidence. (a) Hours of sleep per night and student exam performance. (b) Sales of sunscreen and number of drownings, weekly. (c) Daily temperature in London and your favourite football team's wins. (d) Number of fire trucks at a fire and damage caused.

**Effect of changing units.** A regression of weight (kg) on height (m) gives $W = 50 H + 5$. (a) Predict the weight of a 1.7 m person. (b) If height is now measured in cm instead of m, what is the new regression equation? (c) Verify your equation gives the same prediction at height 170 cm.

**Time-series correlation.** Over 30 years, global $CO_2$ levels and average global temperature both increased. Linear regression gives $r = 0.95$. (a) Does this prove $CO_2$ causes warming? (b) What other evidence would strengthen a causal claim? (c) Could the correlation be coincidence?

**Inverse trend.** A scatter plot has $r = -0.85$. (a) Describe the relationship. (b) The LOBF passes through $(\bar{x}, \bar{y}) = (10, 20)$ with gradient $-2$. State its equation. (c) Predict $y$ at $x = 6$.

**Reading a scatter.** Estimate the correlation coefficient $r$ for each described scatter: (a) Points lie on a perfect straight line going up. (b) Points form a cloud with no trend. (c) Most points cluster around a line going down, but with notable scatter. (d) Points lie on a perfect curve (parabola), symmetric.

**Predict and assess.** A regression of car price (£) on age (years) is $P = -500a + 8000$ for cars aged 0–10 years. (a) Predict price at age 0 and at age 10. (b) Predict price at age 30. Is this reliable? (c) Below what age would the model predict zero price? Is this realistic?

**Comparing methods.** Two students fit lines of best fit to the same scatter plot of 8 points. - Anya draws the line by eye. - Bea uses a calculator to compute the least-squares regression line. Compare and contrast the two approaches: accuracy, repeatability, suitability.

**Choosing a model.** A scatter plot of plant height (cm) vs days since planting (days) has the following data: | Days | 5 | 10 | 15 | 20 | 30 | 50 | 80 | |--------|----|----|----|----|----|----|----| | Height | 2 | 6 | 12 | 20 | 32 | 50 | 65 | (a) Plot the points (sketch). (b) Is a linear model appropriate? Justify by considering the shape of the data. (c) Describe how the rate of growth changes with time.