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Problem-solving Pack
MathematicsYear 10 · Bivariate Statistics
Problem-solving Pack
Name: _________________________________
Date: _________________ Class: ___________
These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.
1Problem 1 of 12
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 and .
(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?
| 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 and .
(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?
Working space
2Problem 2 of 12
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 ().
(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?
(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?
Working space
3Problem 3 of 12
Predicting temperature. A dataset of altitude (m) and temperature (°C) at noon yields the regression line
(a) Interpret the gradient and intercept.
(b) Predict at altitude 1500 m.
(c) Predict at altitude 9000 m (Everest). Is this reliable?
(a) Interpret the gradient and intercept.
(b) Predict at altitude 1500 m.
(c) Predict at altitude 9000 m (Everest). Is this reliable?
Working space
4Problem 4 of 12
Identifying the outlier. Eight data points :
(a) Identify the outlier.
(b) Compute the LOBF gradient with and without the outlier (informally).
(c) Comment on the effect of the outlier.
(a) Identify the outlier.
(b) Compute the LOBF gradient with and without the outlier (informally).
(c) Comment on the effect of the outlier.
Working space
5Problem 5 of 12
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.
(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.
Working space
6Problem 6 of 12
Effect of changing units. A regression of weight (kg) on height (m) gives .
(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.
(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.
Working space
7Problem 7 of 12
Time-series correlation. Over 30 years, global levels and average global temperature both increased. Linear regression gives .
(a) Does this prove causes warming?
(b) What other evidence would strengthen a causal claim?
(c) Could the correlation be coincidence?
(a) Does this prove causes warming?
(b) What other evidence would strengthen a causal claim?
(c) Could the correlation be coincidence?
Working space
8Problem 8 of 12
Inverse trend. A scatter plot has .
(a) Describe the relationship.
(b) The LOBF passes through with gradient . State its equation.
(c) Predict at .
(a) Describe the relationship.
(b) The LOBF passes through with gradient . State its equation.
(c) Predict at .
Working space
9Problem 9 of 12
Reading a scatter. Estimate the correlation coefficient 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.
(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.
Working space
10Problem 10 of 12
Predict and assess. A regression of car price (£) on age (years) is 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?
(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?
Working space
11Problem 11 of 12
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.
- 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.
Working space
12Problem 12 of 12
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.
| 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.
Working space