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Correlation and regression questions
A scatter graph shows whether two variables move together, the product moment correlation coefficient puts a number on how closely, and the regression line turns that relationship into predictions. The skill worth having is knowing what each of those supports as a conclusion and what neither of them can establish.
7 original questions · 19 marks · the correlation and regression notes · Statistics
Every question here is written for this library rather than taken from a past paper. Write your answer out before opening the worked one: the answers award marks point by point, and the marks are easier to see when you have something of your own to compare against.
A scatter diagram gives a product moment correlation coefficient of r = −0.9. Describe the correlation, and sketch the diagram.
Worked 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. The sign gives the direction and the closeness to ±1 gives the strength, so both words are needed.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.
Worked 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. Phrasing earns these marks. Write 'associated with' rather than 'causes', and name hours and marks rather than x and y.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.
Worked 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. That is all the marks claim. Interpolation is not a guarantee. 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.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.
Worked 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. The marks are for the refusal together with the reason, so name extrapolation and quote the range.Ice cream sales and drowning incidents are strongly positively correlated across the months of a year. Explain what this does and does not show.
Worked 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 lurking third variable has manufactured a correlation between two things that never touch. B1 for the association, B1 for ruling out causation, B1 for naming a plausible third variable. Correlation is not causation, and an exam answer has to name a plausible third variable to score in full.A coefficient of r = 0.05 is reported for a large scatter of points. State what this value says.
Worked 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. The coefficient measures straightness and nothing else.The number of bacteria v in a culture after t hours is modelled by v = abt, 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.
Worked 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 = 100.8 = 6.31 and b = 100.12 = 1.32. At t = 5, log10v = 0.8 + 0.6 = 1.4, so v = 101.4 = 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 into v = 6.31 × 1.325 gives the same 25.1, which is a sound check. Marks are lost by matching the coefficients to a and b directly instead of to their logarithms. The straight line lives in log space, so every constant read off it has to be raised as a power of 10 before it means anything about v. In context, b = 1.32 says the culture grows by about 32% each hour.
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
Practise correlation and regression one question at a time
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