Answer Key
Bivariate Statistics
Pack A — Answers
| # | Question | Answer |
|---|---|---|
| 1 | A scatter plot shows that as $x$ increases, $y$ tends to $increase$. Describe the correlation. | Positive correlation |
| 2 | A scatter plot shows random points with no clear trend. Describe the correlation. | No correlation (or zero) |
| 3 | Points lie tightly along a straight line going up. Describe the strength and direction of correlation. | Strong positive correlation |
| 4 | A study plots height (cm) against weight (kg). Which is the (a) independent variable and (b) dependent variable, by convention if predicting weight? | (a) height (x-axis) (b) weight (y-axis) |
| 5 | A graph shows hours of sunshine vs ice cream sales. The points trend upward. Describe the relationship. | Positive correlation: more sunshine, more sales |
| 6 | A line of best fit has equation $y = 3x + 2$. Predict $y$ when $x = 5$. | 17 |
| 7 | A line of best fit has equation $y = 4x + 7$. State the gradient and $y$-intercept. | Gradient 4; y-intercept 7 |
| 8 | A study finds a positive correlation between ice cream sales and shark attacks. Does eating ice cream cause shark attacks? | No — both depend on a third variable (hot weather) |
| 9 | In a scatter of test scores vs revision hours, one point sits well above the line of best fit. What does this represent? | A student who scored higher than predicted for their revision hours |
| 10 | A regression line of $y$ on $x$ is $y = 0.8 x + 5$. Predict $y$ for $x = 20$. | $y = 21$ |
| 11 | A line of best fit passes through $(0, 3)$ and $(5, 18)$. Find its equation. | $y = 3x + 3$ |
| 12 | A regression line for height ($y$, cm) on age ($x$, years) is $y = 6x + 70$. Predict the height of a 9-year-old. | 124 cm |
| 13 | A regression line of test mark ($y$) on hours of revision ($x$) is $y = 5x + 40$. Interpret the gradient in context. | Each extra hour of revision adds 5 marks (on average) |
| 14 | A regression line of test mark ($y$) on hours of revision ($x$) is $y = 5x + 40$. Interpret the $y$-intercept in context. | Predicted mark with zero hours of revision = 40 |
| 15 | For each correlation coefficient $r$, describe the correlation. $r = 0.85$. | Strong positive correlation |
| 16 | A line of best fit is $y = 4x + 6$. Find $x$ when $y = 30$. | $x = 6$ |
| 17 | A regression line of $y$ on $x$ passes through the mean point $(\bar{x}, \bar{y})$. If $\bar{x} = 5$ and the line is $y = 3x + 4$, find $\bar{y}$. | $\bar{y} = 19$ |
| 18 | A regression equation was fitted using $x$-values from 5 to 20. Using the line, $y$ is predicted for $x = 10$ (case A) and $x = 30$ (case B). Which is interpolation, which is extrapolation? | $x = 10$: interpolation; $x = 30$: extrapolation |
| 19 | Plot 1 has $r = 0.9$ and Plot 2 has $r = 0.2$. Which shows a stronger linear relationship? | Plot 1 ($r$ closer to 1) |
| 20 | A scatter plot of time spent gaming vs test scores has line of best fit $y = -2x + 70$. Interpret in context. | Each hour of gaming predicts a 2-mark decrease; with zero gaming the predicted score is 70 |
| 21 | Five data points: (1, 3), (2, 5), (3, 7), (4, 9), (5, 11). Find the regression line by inspection. | $y = 2x + 1$ |
| 22 | A regression line $y = 0.5x + 20$ was fitted to $x$-values from 10 to 50. (a) Predict $y$ at $x = 30$. (b) At $x = 80$ the prediction would be? | (a) 35 (b) 60 (extrapolation — unreliable) |
| 23 | A car's mileage (km) vs its resale value (£). Suggested data has $r \approx -0.92$. Interpret. | Strong negative: more mileage → lower value |
| 24 | A regression line has a strong outlier. Will removing it (a) increase or decrease $|r|$? (b) Why? | (a) Increase $|r|$ — removes scatter (b) Outliers reduce correlation strength |
| 25 | Five data points: $\bar{x} = 5, \bar{y} = 12$; regression line has gradient $2$. State the regression equation. | $y = 2x + 2$ |
| 26 | Categorise the correlation based on $r$: (a) $r = 0.3$, (b) $r = 0.7$, (c) $r = -0.95$, (d) $r = 0.05$. | (a) Weak positive (b) Strong positive (c) Very strong negative (d) None |
| 27 | Five data points have $\bar{x} = 4$ and $\bar{y} = 11$. Four are $(2, 7), (3, 9), (5, 13), (6, 15)$. Find the fifth. | $(4, 11)$ |
| 28 | Studies find $r = 0.9$ between countries' chocolate consumption and number of Nobel Prize winners. Does eating chocolate cause Nobel Prizes? | No — confounding variable (national wealth) drives both |
| 29 | LOBF $y = 2x + 1$ predicts $y$ at $x = 5$. Actual measured value is 9. What is the residual? | Residual = $-2$ (observed minus predicted) |
| 30 | A regression line predicts temperature in °C from altitude (m): $T = -0.006a + 15$. Predict $T$ at altitude 2000 m. | $T = 3°$C |
| 31 | A regression has $r = 0.8$. What proportion of variation in $y$ is explained by $x$? | 64% ($r^2 = 0.64$) |
| 32 | Two studies of "$y$ vs $x$" have regression lines: Study A: $y = 3x + 5$, $r = 0.6$. Study B: $y = 3x + 5$, $r = 0.9$. Which is a better fit? Why? | B is better fit (higher $|r|$) |
| 33 | A regression line $y = 2x + 5$ has $r^2 = 0.7$. At $x = 10$, you predict $y = 25$. What does $r^2 = 0.7$ tell you? | 70% of variation explained — moderate-strong predictive value |
| 34 | Three points and their LOBF predictions: $(1, 4) \to$ predicted 3.5; $(2, 6) \to$ predicted 5.5; $(3, 8) \to$ predicted 7.5. Find the sum of residuals. | 1.5 |
| 35 | Five students rank their preference for two subjects. The rank correlation is 0.8. Interpret. | Strong positive rank agreement |
| 36 | A scatter plot looks like a U-shape. A student fits a straight LOBF and finds $r = 0$. What does this tell us about the relationship? | There IS a relationship — but not linear |
| 37 | A regression line of "ice cream sales" on "temperature (°C)" gives $y = 2.5x + 5$ for $x$ in $[15, 35]$. At $x = -10$°C, the prediction is $-20$. Comment. | Extrapolation — and negative sales is nonsensical |
| 38 | A dataset has $r = -0.7$ and $r^2 = 0.49$. Write a one-sentence summary for a non-technical audience. | There is a fairly strong negative relationship; the model explains about 49% of the variation |
| 39 | For a sample of cars, fuel economy ($y$, mpg) vs engine size ($x$, L) gives $y = -8x + 50$. (a) Predict mpg for a 2 L engine. (b) Comment on $x = 5$ L. | (a) 34 mpg (b) 10 mpg — likely extrapolation |
| 40 | A study finds $r = 0.95$ between heights of parent and child. Does this mean a tall parent guarantees a tall child? | No — strong tendency, not certainty |
Pack B — Answers
| # | Question | Answer |
|---|---|---|
| 1 | A scatter plot shows that as $x$ increases, $y$ tends to $decrease$. Describe the correlation. | Negative correlation |
| 2 | A scatter plot shows random points with no clear trend. Describe the correlation. | No correlation |
| 3 | Points lie tightly along a straight line going up. Describe the strength and direction of correlation. | Strong negative correlation |
| 4 | A study plots height (cm) against weight (kg). Which is the (a) independent variable and (b) dependent variable, by convention if predicting weight? | (a) hours studied (b) test score |
| 5 | A graph shows hours of sunshine vs ice cream sales. The points trend upward. Describe the relationship. | Positive correlation: more revision, higher score |
| 6 | A line of best fit has equation $y = 2x + 10$. Predict $y$ when $x = 8$. | 26 |
| 7 | A line of best fit has equation $y = -2x + 15$. State the gradient and $y$-intercept. | Gradient $-2$; y-intercept 15 |
| 8 | A study finds a positive correlation between ice cream sales and shark attacks. Does eating ice cream cause shark attacks? | No — both depend on size of fire |
| 9 | In a scatter of test scores vs revision hours, one point sits well above the line of best fit. What does this represent? | A student who scored lower than predicted |
| 10 | A regression line of $y$ on $x$ is $y = 0.8 x + 5$. Predict $y$ for $x = 20$. | $y = 21$ |
| 11 | A line of best fit passes through $(0, 4)$ and $(6, 22)$. Find its equation. | $y = 3x + 4$ |
| 12 | A regression line for height ($y$, cm) on age ($x$, years) is $y = 6x + 70$. Predict the height of a 9-year-old. | 120 cm |
| 13 | A regression line of test mark ($y$) on hours of revision ($x$) is $y = 5x + 40$. Interpret the gradient in context. | Each extra hour of revision adds 8 marks (on average) |
| 14 | A regression line of test mark ($y$) on hours of revision ($x$) is $y = 5x + 40$. Interpret the $y$-intercept in context. | Predicted mark with zero revision = 50 |
| 15 | For each correlation coefficient $r$, describe the correlation. $r = -0.42$. | Moderate negative correlation |
| 16 | A line of best fit is $y = 3x + 5$. Find $x$ when $y = 32$. | $x = 9$ |
| 17 | A regression line of $y$ on $x$ passes through the mean point $(\bar{x}, \bar{y})$. If $\bar{x} = 8$ and the line is $y = 2x + 6$, find $\bar{y}$. | $\bar{y} = 22$ |
| 18 | A regression equation was fitted using $x$-values from 5 to 20. Using the line, $y$ is predicted for $x = 10$ (case A) and $x = 30$ (case B). Which is interpolation, which is extrapolation? | $x = 4$: interpolation; $x = 20$: extrapolation |
| 19 | Plot 1 has $r = 0.9$ and Plot 2 has $r = 0.2$. Which shows a stronger linear relationship? | Plot 1 ($|r| = 0.85$ vs 0.1) |
| 20 | A scatter plot of time spent gaming vs test scores has line of best fit $y = -2x + 70$. Interpret in context. | Each hour of gaming predicts a 1.5-mark decrease; intercept 80 |
| 21 | Five data points: (1, 3), (2, 5), (3, 7), (4, 9), (5, 11). Find the regression line by inspection. | $y = 3x + 4$ |
| 22 | A regression line $y = 0.5x + 20$ was fitted to $x$-values from 10 to 50. (a) Predict $y$ at $x = 30$. (b) At $x = 80$ the prediction would be? | (a) 40 (b) 70 (extrapolation) |
| 23 | A car's mileage (km) vs its resale value (£). Suggested data has $r \approx -0.92$. Interpret. | Strong positive: older athletes have slower sprints |
| 24 | A regression line has a strong outlier. Will removing it (a) increase or decrease $|r|$? (b) Why? | (a) Possibly decrease $|r|$ (b) An aligned outlier may anchor a strong correlation |
| 25 | Five data points: $\bar{x} = 5, \bar{y} = 12$; regression line has gradient $3$. State the regression equation. | $y = 3x - 3$ |
| 26 | Categorise the correlation based on $r$: (a) $r = 0.3$, (b) $r = 0.7$, (c) $r = -0.95$, (d) $r = 0.05$. | (a) Weak negative (b) Strong positive (c) None (d) Moderate positive |
| 27 | Five data points have $\bar{x} = 4$ and $\bar{y} = 11$. Four are $(2, 7), (3, 9), (5, 13), (6, 15)$. Find the fifth. | $(5, 14)$ |
| 28 | Studies find $r = 0.9$ between countries' chocolate consumption and number of Nobel Prize winners. Does eating chocolate cause Nobel Prizes? | No — population size drives both |
| 29 | LOBF $y = 2x + 1$ predicts $y$ at $x = 5$. Actual measured value is 9. What is the residual? | Residual = 0 |
| 30 | A regression line predicts temperature in °C from altitude (m): $T = -0.006a + 15$. Predict $T$ at altitude 2000 m. | $T = -6°$C |
| 31 | A regression has $r = 0.8$. What proportion of variation in $y$ is explained by $x$? | 36% |
| 32 | Two studies of "$y$ vs $x$" have regression lines: Study A: $y = 3x + 5$, $r = 0.6$. Study B: $y = 3x + 5$, $r = 0.9$. Which is a better fit? Why? | B is better fit |
| 33 | A regression line $y = 2x + 5$ has $r^2 = 0.7$. At $x = 10$, you predict $y = 25$. What does $r^2 = 0.7$ tell you? | 40% of variation explained — weak predictive value |
| 34 | Three points and their LOBF predictions: $(1, 4) \to$ predicted 3.5; $(2, 6) \to$ predicted 5.5; $(3, 8) \to$ predicted 7.5. Find the sum of residuals. | 1.5 |
| 35 | Five students rank their preference for two subjects. The rank correlation is 0.8. Interpret. | Moderate negative rank agreement |
| 36 | A scatter plot looks like a U-shape. A student fits a straight LOBF and finds $r = 0$. What does this tell us about the relationship? | Strong non-linear relationship hidden by $r$ |
| 37 | A regression line of "ice cream sales" on "temperature (°C)" gives $y = 2.5x + 5$ for $x$ in $[15, 35]$. At $x = -10$°C, the prediction is $-20$. Comment. | Same — extrapolation gives nonsense |
| 38 | A dataset has $r = -0.7$ and $r^2 = 0.49$. Write a one-sentence summary for a non-technical audience. | There is a moderate positive relationship; the model explains about 25% of the variation |
| 39 | For a sample of cars, fuel economy ($y$, mpg) vs engine size ($x$, L) gives $y = -8x + 50$. (a) Predict mpg for a 2 L engine. (b) Comment on $x = 5$ L. | (a) 45 mpg (b) 5 mpg — extrapolation |
| 40 | A study finds $r = 0.95$ between heights of parent and child. Does this mean a tall parent guarantees a tall child? | No — moderate tendency only |
Problems — Worked Solutions
**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?
(a) Strong positive (b) $\bar{x} = 7.7$, $\bar{y} = 61.5$ (c) Slope ≈ 4.3; $y \approx 4.3x + 28$ (d) ≈ 45.4 (interpolation, reasonable)
**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?
No — coincidence / lurking variables. The lesson: correlation does not imply causation.
**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?
(a) Each 1 m up loses 0.0065°C; sea level baseline 15°C (b) 5.25°C (c) ≈ −43.5°C — probably extrapolation
**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.
(a) $(7, 25)$ is the outlier (b) With: ≈ 2.5; Without: ≈ 2.0 (c) Outlier inflates gradient and reduces $r$
**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) Plausibly causal (sleep affects cognition) (b) Confounding (hot weather) (c) Coincidence (d) Confounding (size of fire)
**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.
(a) 90 kg (b) $W = 0.5 H + 5$ (c) ✓ — same 90 kg
**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?
(a) No, alone (b) Physical mechanism, experiments, models (c) Unlikely given mechanism evidence
**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 $x = 6$.
(a) Strong negative (b) $y = -2x + 40$ (c) $y = 28$
**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.
(a) $r = 1$ (b) $r ≈ 0$ (c) $r$ around $-0.7$ (d) $r ≈ 0$
**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?
(a) £8000 and £3000 (b) £-7000 — nonsensical and extrapolation (c) 16 years — partly realistic (very old cars near 0 value)
**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.
See working — Bea's method is more accurate and repeatable.
**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.
(a) See sketch (b) No — growth slows over time (c) Initially rapid, then slows (sub-linear)