Correlation

(a) Negative correlation: older cars are worth less. Age is the explanatory variable, so it goes on the $x$-axis.

(b) Positive correlation: taller people tend to have larger feet. Height on the $x$-axis.

(c) No correlation: the two are unrelated. (Either could go on the $x$-axis, since neither explains the other.)

(a) $y = 12.5 + 3.4(6) = 32.9$.

(b) For each increase of 1 in $x$, $y$ increases by about 3.4.

(a) Interpolation, since 5 lies inside the range 2 to 15. This prediction can be trusted (as far as the model fits).

(b) Extrapolation, since 20 is outside the range. There is no evidence the pattern continues that far, so it is unreliable.

Correlation does not imply causation. A third variable — hot, sunny weather — increases both ice cream sales and the time people spend outside getting sunburnt. The two rise together without either causing the other.

(a) For each 1 °C rise in midday temperature, the café sells about 4.9 more cold drinks.

(b) $s = 26.2 + 4.88(19) = 118.9$, so about 119 drinks. This is interpolation, so it should be reasonably reliable.

(c) $s = 26.2 + 4.88(35) = 197$, so about 197 drinks. But 35 °C is far outside the recorded range of 12–26 °C, so this is extrapolation. Sales might level off once the café runs out of stock or seats, so the estimate is unreliable.

(a) The value falls by about £1750 for each extra year of age. The intercept suggests a new car (age 0) would be worth £14 500, though no car in the sample was new, so that is an extrapolation.

(b) 12 years is outside the range 1 to 9 years. The model would predict $14\,500 - 1750(12) = -£6500$, a negative value, which is impossible: the linear pattern clearly does not continue.

(a) The second cluster lies to the right of the first but lower down. A single line mostly measures the difference between the two groups rather than the pattern within them, so it slopes the opposite way.

(b) The data should be separated, with a regression line fitted to each group, and the two groups described separately. The scatter should always be looked at, not just the equation of the line.

(a) Strong positive correlation: longer fish tend to be heavier.

(b) Length. A fish’s mass depends on its size, rather than the other way round, so length goes on the $x$-axis as the explanatory variable.

(c) It would sit well above the pattern: at 24 cm the other fish weigh around 1.5 kg, so 3.9 kg is far heavier than expected. It should be investigated as a possible recording error before being included. If it is genuine, it should be kept and commented on, since it may be a different species.

(a) Each extra hour of study per week is associated with about 1.06 more marks.

(b) $y = 37.4 + 1.06(14) = 52.2$, so about 52 marks (interpolation, so reasonably reliable).

(c) $y = 37.4 + 1.06(40) = 79.8$, so about 80 marks. Two reasons to distrust it: 40 hours is far outside the range 5–28 hours (extrapolation), and the relationship is unlikely to stay linear — extra hours probably have less and less effect, and tiredness may reduce performance.

(d) The data show association, not cause. More able or better-motivated students may both study more and score higher, so a third factor could explain the link. Only a controlled experiment could establish cause.

(a) The wealth of the town. Wealthier areas can fund more libraries and tend to have better-resourced schools, which raises results.

(b) If wealth is the real driver, building libraries alone may not improve results at all, and the money might be better spent directly on schools.

(c) A controlled study: compare similar towns where libraries are added with ones where they are not, over time, keeping other factors as alike as possible.

(a) The regression line of $y$ on $x$ is chosen to make the vertical distances from the points small, so it is designed for predicting $y$ from $x$. Rearranging it to predict $x$ from $y$ is not the same as the line of $x$ on $y$, which would be fitted the other way round, and it generally gives a different answer.

(b) If the correlation is very strong, the two lines are close together, so the estimate is reasonable — provided $y = 31$ also lies inside the range of the observed $y$ values.

(a) A straight line fits the first 8 weeks well, but the later points curve away from it. Over the whole 12 weeks, a linear model is not suitable: it will overestimate the later heights.

(b) Week 20 is outside the data (extrapolation), and the pattern was already bending away from the line by week 12, so the linear trend has clearly stopped. The prediction of $2.1 + 3.4(20) = 70.1$ cm is almost certainly far too high.

(c) Use a model that levels off, since plants approach a maximum height — for example fitting a curve, or a straight line only to the first 8 weeks while stating that it applies to that period alone.