4.4 Correlation and regression questions for IB Maths AI

65 questions on Correlation and regression, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 4 Statistics and probability
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	20	6	26
Paper 2	25	12	37
Paper 3HL only	0	2	2
Every paper	45	20	65

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00627Paper 1 · HL only · 7 marks

The following data were collected in a physics experiment.

$x$	1	2	3	4	5	6	8
$y$	3.1	11.8	27.5	46.0	74.9	105.2	190.0

Lucas wants to know whether the relationship between $x$ and $y$ is better modelled by an exponential function $y = k a^{x}$ or by a power function $y = c x^{n}$ .

(a)
Find Pearson's product-moment correlation coefficient for
(i) $x$ and $\ln y$ ;
(ii) $\ln x$ and $\ln y$ .
2 marks
(b)
State, with a reason, which type of function is the better model.
2 marks
(c)
Find the value of $c$ and the value of $n$ .
3 marks

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Question 2 · AI-00203Paper 2 · 13 marks

Maya runs a café at a beach. For ten days in summer she records the maximum temperature, $T$ °C, and the number of iced drinks sold, $n$ .

$T$ (°C)	24	27	29	31	22	33	26	2.9	30	28
$n$	129	138	179	176	107	198	150	168	164	162

(a)
One of the temperatures has been recorded in error. Identify this value and suggest what the temperature probably was.
2 marks
(b)
If the error is not noticed, the value of Pearson's product-moment correlation coefficient for all ten days is $r = 0.230$ . Maya removes the day with the error.
Find the value of $r$ for the remaining nine days.
2 marks
(c)
Comment on the effect of the error on the correlation coefficient.
1 mark
(d)
For the nine days, find the equation of the regression line of $n$ on $T$ .
2 marks
(e)
Interpret, in context, the gradient of the regression line.
1 mark
(f)
Use your regression equation to estimate the number of iced drinks Maya sells on a day when the maximum temperature is 25 °C.
2 marks
(g)
Maya wants to use the equation to estimate sales on a winter day with a maximum temperature of 12 °C. Explain why this estimate may not be reliable.
2 marks
(h)
Maya says the café was closed on one day during the ten-day period, so no data were recorded. She suggests recording 0 drinks for that day. Explain why this would be inappropriate.
1 mark

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Question 3 · AI-01461Paper 3 · HL only · 27 marks

The purpose of this question is to model the cooling of bread so that a bakery knows when its loaves can be sliced.

At Brot und Mehl bakery, loaves are taken out of the oven with a core temperature of 96 °C and left on racks in a cooling room kept at 22 °C. Newton's law of cooling states that the rate at which the temperature of an object decreases is proportional to the difference between its temperature and the temperature of its surroundings.

Let $T$ °C be the core temperature of a loaf $t$ minutes after it is taken out of the oven.

(a)
Write down a differential equation for $T$ , using a positive constant $k$ .
2 marks
(b)
By solving your differential equation, show that $T = 22 + A e^{- k t}$ , where $A$ is a constant.
4 marks
(c)(i)
Write down the value of $A$ .
1 mark
(c)(ii)
After 20 minutes the core temperature of a loaf is 61 °C.
Show that $k = 0.0320$ , correct to three significant figures.
2 marks
(d)
A loaf can be sliced once its core temperature has fallen to 35 °C.
Use the model to find how long after leaving the oven a loaf can be sliced. Give your answer to the nearest minute.
2 marks
(e)
Find the rate at which the core temperature is changing at the moment the loaf leaves the oven.
2 marks
(f)(i)
The head baker, Jonas, thinks that $k$ should be estimated from more than one reading. He records the core temperature of a loaf every 10 minutes.
$t$ (minutes)
0
10
20
30
40
50
60
$T$ (°C)
96
75
61
51
43
37
33
$y = \ln (T - 22)$
4.304
3.664
3.367
2.708
Complete the table, giving each value of $y$ correct to three decimal places.
2 marks
(f)(ii)
Find the equation of the regression line of $y$ on $t$ .
2 marks
(f)(iii)
Hence find Jonas's estimates for $k$ and for $A$ .
3 marks
(g)
Write down the value of the coefficient of determination, $R^{2}$ , for the regression in part (f)(ii), and interpret it in context.
2 marks
(h)(i)
On busy days the ovens warm the cooling room. Jonas models the room temperature as $22 + 0.15 t$ °C and uses the differential equation
$\frac{d T}{d t} = - 0.032 (T - 22 - 0.15 t), T (0) = 96.$
Use Euler's method with a step length of 5 minutes to estimate the core temperature of a loaf after 40 minutes on a busy day.
4 marks
(h)(ii)
On a busy day, the measured core temperature of a loaf after 40 minutes is 45.2 °C. Suggest how Jonas could use Euler's method to obtain a more accurate estimate.
1 mark

$t$ (minutes)	0	10	20	30	40	50	60
$T$ (°C)	96	75	61	51	43	37	33
$y = \ln (T - 22)$	4.304		3.664	3.367		2.708

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Question 4 · AI-01007Paper 1 · 6 marks

Inês runs a café in Porto. On seven days in June she records the maximum daily temperature, $T$ °C, and the number of iced coffees sold, $n$ .

$T$ (°C)	18	21	23	25	27	30	32
$n$	42	58	57	71	69	90	88

(a)
Find Pearson's product-moment correlation coefficient, $r$ , for these data.
2 marks
(b)
Write down the equation of the regression line of $n$ on $T$ .
2 marks
(c)
Use your equation to estimate the number of iced coffees sold on a day when the maximum temperature is 28 °C.
2 marks

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Question 5 · AI-00265Paper 2 · 15 marks

A weather station beside Lake Bled records the mean July temperature, $T$ °C, every second year. The data are shown in the following table, where $x$ is the number of years after 2010.

$x$ (years after 2010)	0	2	4	6	8	10	12
$T$ (°C)	18.2	18.5	18.4	19.0	19.1	19.6	19.8

Matej models the data with the straight line that passes through the first and last data points.

(a)
Find the gradient of Matej's line.
2 marks
(b)
Interpret the gradient in the context of the question, stating its units.
2 marks
(c)
Write down the equation of Matej's line in the form $T = m x + c$ .
1 mark
(d)
Use Matej's line to estimate the mean July temperature in 2025.
2 marks
(e)
Use your graphic display calculator to find
(i) Pearson's product-moment correlation coefficient, $r$ ;
(ii) the equation of the regression line of $T$ on $x$ .
4 marks
(f)
Use the regression line to estimate the mean July temperature in 2017.
2 marks
(g)
Matej's sister claims that the regression line can be used to predict the mean July temperature in 2060. Comment on her claim.
2 marks

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Question 6 · AI-01644Paper 3 · HL only · 25 marks

This question compares models for the power output of a wind turbine.

An engineer, Tomasz, records the wind speed, $v$ m s⁻¹, and the power output, $P$ kW, of a small wind turbine at a coastal site.

$v$ (m s⁻¹)	4	5	6	7	8	9	10	11	12
$P$ (kW)	58	102	190	285	441	612	862	1120	1475

(a)(i)
Find the equation of the regression line of $P$ on $v$ .
2 marks
(a)(ii)
Write down the coefficient of determination, $R^{2}$ , for this linear model.
1 mark
(a)(iii)
Calculate the residuals of the linear model at $v = 4$ , $v = 8$ and $v = 12$ , and hence explain why a linear model is not appropriate, despite the high value of $R^{2}$ .
3 marks
(b)(i)
Tomasz next considers a power model $P = a v^{b}$ .
Show that this model gives a linear relationship between $\ln P$ and $\ln v$ .
2 marks
(b)(ii)
Use your graphic display calculator to find the values of $a$ and $b$ .
2 marks
(b)(iii)
For a model $P = f (v)$ , the sum of square residuals is $S S_{r e s} = \sum_{i = 1}^{9} (P_{i} - f (v_{i}))^{2}$ .
Find the value of $S S_{r e s}$ for the power model.
2 marks
(c)(i)
Physical theory suggests that the power output is proportional to the cube of the wind speed, so Tomasz now considers the model $P = k v^{3}$ . He chooses $k$ to minimize
$S (k) = \sum_{i = 1}^{9} {(P_{i} - k v_{i}^{3})}^{2} .$
Show that $S (k)$ is minimized when $k = \frac{\sum v_{i}^{3} P_{i}}{\sum v_{i}^{6}}$ .
4 marks
(c)(ii)
Hence find the value of $k$ .
2 marks
(d)
For the model $P = k v^{3}$ , $S S_{r e s} = 516$ , correct to three significant figures.
Tomasz notices that this one-parameter model has a smaller value of $S S_{r e s}$ than the power model in part (b), which has two parameters. Explain why this is possible, and state which of the two models fits the data better.
3 marks
(e)
On one day, the wind speed at the site $t$ hours after midnight is modelled by $v = 7 + 3 \sin (\frac{π t}{12})$ , for $0 \leq t \leq 24$ .
The energy, in kWh, produced by the turbine in one day is $\int_{0}^{24} P d t$ . Use the model $P = k v^{3}$ to find the energy produced that day.
3 marks
(f)
The turbine's maximum power output is 1500 kW. Comment on using the model $P = k v^{3}$ to predict the power output when the wind speed is 15 m s⁻¹.
1 mark

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