Questions on correlation and regression

Question 1

A scatter diagram gives a product moment correlation coefficient of r = −0.9. Describe the correlation, and sketch the diagram.

Accepted Answer

Strong negative correlation. As one variable rises the other falls, and the points sit close to a downhill straight line. B1 for the direction, B1 for the strength.

Question 2

A regression line for exam mark y against revision hours x is y = 3.5 + 0.6x. Interpret the numbers 3.5 and 0.6 in context.

Accepted Answer

The gradient 0.6 says that each extra hour of revision is associated with about 0.6 more marks. The intercept 3.5 says that a student who did no revision is predicted about 3.5 marks. B1 for the gradient in context, B1 for the intercept in context. Write 'associated with' rather than 'causes', and name hours and marks rather than x and y.

Question 3

A scatter diagram is drawn for two variables. State which of the two is plotted on the horizontal axis, and give the name used for each of the pair.

Accepted Answer

The explanatory variable goes on the horizontal axis and the response variable on the vertical axis. B1 for the explanatory variable on the horizontal axis, B1 for the response variable on the vertical. The explanatory variable, also called the independent variable, is the one chosen or set; the response, or dependent, variable is the one measured in reply to it. Which is which decides the direction a fitted line may be read in, since a line of y on x predicts the response from the explanatory variable and not the reverse.

Question 4

One scatter diagram gives r = 0.98 and a second gives r = −0.98. State what the two diagrams have in common, state how they differ, and state what a value of r = 1 would mean.

Accepted Answer

Both are strong. The distance of r from zero is 0.98 in each case, so in both diagrams the points sit close to a straight line. B1 for the shared strength. They differ in direction: r = 0.98 is positive, so the points rise from left to right, while r = −0.98 is negative and they fall. B1 for the direction. A value of r = 1 would mean every point lies exactly on one straight line of positive gradient, with no scatter about it at all. B1.

Question 5

A regression line for exam mark y against revision hours x is y = 3.5 + 0.6x. Predict the mark of a student who revised for 10 hours, given that the data covered 2 to 15 hours.

Accepted Answer

y = 3.5 + 0.6 × 10 = 9.5 marks. Since 10 hours lies inside the observed range of 2 to 15 hours this is interpolation, so the prediction is not exposed to the extra risk of extrapolation and the data do at least speak to that part of the line. M1 for the substitution, A1 for 9.5 marks identified as interpolation. How much the number is worth also depends on how well a straight line fits, how tightly the points hug it, and how the data were collected. A weak r or an obvious curve would undermine 9.5 even at 10 hours. Note also that this line predicts y from x only. To go the other way and estimate the hours behind a known mark you would need the regression line of x on y, which is a different line.

Question 6

A teacher uses the line y = 3.5 + 0.6x, fitted to revision hours from 2 to 15, to predict the mark after 40 hours of revision. Comment on the prediction.

Accepted Answer

The arithmetic gives 3.5 + 24 = 27.5 marks, and the number deserves no trust. 40 hours lies far outside the 2 to 15 hour window the line was fitted to, so this is extrapolation. Nothing in the data shows the straight-line pattern continues that far out, and in practice the returns to revision would flatten. B1 for 27.5, B1 for naming extrapolation, B1 for quoting the fitted range.

Question 7

Ice cream sales and drowning incidents are strongly positively correlated across the months of a year. Explain what this does and does not show.

Accepted Answer

It shows that the two rise and fall together. It does not show that either one causes the other. Hot weather drives both, bringing more swimming and more ice cream, so a third variable produces the correlation even though neither affects the other. B1 for the association, B1 for ruling out causation, B1 for naming a plausible third variable.

Question 8

A coefficient of r = 0.05 is reported for a large scatter of points. State what this value says.

Accepted Answer

There is almost no linear relationship. The points form a shapeless cloud about any straight line. A relationship is not ruled out, though, because a strong curve can sit behind a near-zero r. B1 for almost no linear relationship, B1 for a relationship not being ruled out.

Question 9

A scatter diagram of arm span against height is drawn for a sample containing both young children and adults. The points form two separate clusters with a wide gap between them. Explain why a product moment correlation coefficient calculated from all the points overstates the strength of the relationship inside either group, and state what should be done before any line is fitted.

Accepted Answer

The sample holds two distinct sections of the population rather than one. B1 for identifying the two sections. The coefficient measures how close the whole set of points lies to a single straight line, and the gap between the clusters is itself a strong straight-line pattern: adults sit high on both axes and children low on both, so r comes out large however scattered each cluster is inside itself. B1 for the gap driving the value. A line fitted to everything then describes neither group, because it runs through the empty space between them. B1. The two sections should be separated and each examined on its own, with its own diagram, coefficient and line. B1.

Question 10

The time y minutes taken by a machine to cool is modelled by y = 12.4 − 0.35x, where x is the fan speed setting. The line was fitted to settings from 5 to 40. Find the predicted cooling time at x = 20 and at x = 60, and state which of the two predictions can be relied on.

Accepted Answer

At x = 20, y = 12.4 − 0.35 × 20 = 12.4 − 7 = 5.4 minutes. M1 for the substitution, A1 for 5.4 minutes. At x = 60, y = 12.4 − 0.35 × 60 = 12.4 − 21 = −8.6 minutes. M1 for the second substitution. The first can be relied on: 20 lies inside the fitted window of 5 to 40, so it is interpolation. The second cannot, because 60 lies far outside that window, and the arithmetic returns a negative cooling time, which no machine can take. A1 for accepting the first and rejecting the second with the fitted range quoted. An impossible value is the model saying it has been pushed past its data: a straight line has to meet the axis somewhere, and nothing in the observations says the real relationship does.

Question 11

A regression line of y on x has been fitted to a set of bivariate data. Explain why that line should not be used to estimate a value of x from a given value of y.

Accepted Answer

The line of y on x is fitted by making the vertical distances from the points to the line as small as possible, so it is built to predict the response y from the explanatory variable x. B1 for what the fitting minimises, or for the direction the line is built for. Reading it backwards asks it a question it was not fitted for, and in general returns a different answer from the line that was. B1. Estimating x from y calls for the regression line of x on y, a second and different line through the same points. B1. The two lines coincide only when every point already lies exactly on one straight line.

Question 12

The number of bacteria v in a culture after t hours is modelled by v = ab^t, where a and b are constants. The regression line of log₁₀v on t is log₁₀v = 0.8 + 0.12t. Find the values of a and b, and estimate v when t = 5.

Accepted Answer

Take logs of the model first: log₁₀v = log₁₀a + t log₁₀b. Comparing with the fitted line, log₁₀a = 0.8 and log₁₀b = 0.12, so a = 10^0.8 = 6.31 and b = 10^0.12 = 1.32. At t = 5, log₁₀v = 0.8 + 0.6 = 1.4, so v = 10^1.4 = 25.1 bacteria. M1 for taking logs of the model, A1 for a = 6.31, A1 for b = 1.32, M1 for substituting t = 5, A1 for 25.1. Substituting the unrounded a and b into v = ab⁵ gives the same 25.1, which is a sound check. The coefficients of the fitted line match log a and log b, not a and b themselves. In context, b = 1.32 says the culture grows by about 32% each hour.

Question 13

Two variables x and y are thought to satisfy either y = axⁿ or y = kb^x. A scatter of log₁₀y against log₁₀x lies close to a straight line, while a scatter of log₁₀y against x is clearly curved. State which of the two models fits, with a reason, and given that the fitted line is log₁₀y = 0.6 + 1.5 log₁₀x, find a and n and estimate y when x = 16.

Accepted Answer

Take logs of each model and see which one straightens against which axis. For y = axⁿ, log₁₀y = log₁₀a + n log₁₀x, a straight line against log₁₀x. For y = kb^x, log₁₀y = log₁₀k + x log₁₀b, a straight line against x. The straight scatter is the one against log₁₀x, so the power model y = axⁿ is the one that fits. M1 for taking logs of a model, A1 for choosing y = axⁿ with that reason. Comparing coefficients with the fitted line, log₁₀a = 0.6 and n = 1.5, so a = 10^0.6 = 3.98 and n = 1.5. A1 for a = 3.98, B1 for n = 1.5. At x = 16, log₁₀y = 0.6 + 1.5 log₁₀16 = 0.6 + 1.806 = 2.406, so y = 10^2.406 = 255 to 3 significant figures. M1 for substituting, A1 for 255. Going the other way, y = 3.98 × 16^1.5 = 3.98 × 64 = 255 as well.

Question 14

Fifteen points on a scatter diagram give a product moment correlation coefficient of r = 0.42. A sixteenth point is then added, far to the right of the others and far above them, and r for all sixteen points rises to 0.88. Explain how one point can change r by this much, and state what should be done about it.

Accepted Answer

The coefficient is a property of the whole set of points, measured against the mean of each variable. B1 for r describing the set as a whole. A point lying far out along both axes is a long way from both means, so its contribution to the calculation is large, and it can pull the coefficient towards itself on its own. B1 for the influence of a distant point. The other fifteen points have not moved and still show the weak association they showed before, so 0.88 is describing one observation more than a pattern in the data. B1. The point should first be checked as a possible recording or transcription error. B1. If it survives that check it is a real observation and may not simply be deleted; what is sound is to report r both with and without it, saying which is which. B1.

Question 15

A study of 400 students finds a strong positive correlation between the number of hours slept on a school night and the mark scored in a test the next day. A newspaper reports that sleeping more raises test marks. Give two distinct reasons why that conclusion is unsafe, and describe the evidence that would be needed to support it.

Accepted Answer

First, the causation could run the other way. A student anxious about a test sleeps badly and also scores poorly, so short sleep and low marks would travel together with the test driving the sleep rather than the sleep driving the test. B1 for reverse causation with a mechanism named. Second, a third variable can produce the correlation without either variable touching the other: a settled home routine brings both an early bedtime and time spent working, and would raise both figures at once. B1 for a plausible confounding variable named in context. Observational data cannot separate those three explanations, because the students chose their own bedtimes. B1. Supporting the claim needs an experiment: allocate students at random to sleep schedules, so that the groups differ in sleep and in nothing else systematic, then compare the marks. B1 for random allocation. Even then the finding would apply only to the population sampled and to the range of sleep lengths tested. B1.

Question 16

The scatter diagram shows, for each of ten students, the time x, in hours, spent watching television during one week and the mark y scored in a test at the end of that week. The line drawn on the diagram is the regression line of y on x for nine of the ten points; the remaining point was left out when the line was fitted. Write down the coordinates of the point that was left out, describe the correlation shown by the other nine points, and use the line to find the equation of the line, interpreting the gradient in this context.

Accepted Answer

The point the line takes no account of is the one lying far above the pattern, at (16, 90): a student who watched 16 hours of television and still scored 90, where the line stands at 50. B1 for the coordinates. The other nine points fall steadily from top left to bottom right and lie close to the line, so they show strong negative correlation. B1 for the description. Two points where the line crosses a gridline intersection are (0, 90) and (20, 40), so the gradient is (40 − 90)/(20 − 0) = −50/20 = −2.5. M1 for a gradient taken from two points read off the line, A1 for −2.5. The line meets the vertical axis at 90, so its equation is y = 90 − 2.5x. A1 for the equation. The gradient says that each extra hour of television in the week goes with a fall of 2.5 marks in the test. B1 for the interpretation in context. The gradient is a statement about how the two measurements move together and not about cause: a student who gives more of the week to television has less of it left for revision, and it is the revision the line cannot see.