4.10 Regression of x on y questions for IB Maths AA

14 questions on Regression of x on y, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Analysis and Approaches
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	All
Paper 1	5	5
Paper 2	9	9
Every paper	14	14

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 · AA-00395Paper 1 · 5 marks

A librarian surveys a group of students. For each student, the number of hours spent reading each week, $x$ , and their score in a reading test, $y$ , are recorded.

For these data, the regression line of $y$ on $x$ is $y = 2.5 x + 45$ and the regression line of $x$ on $y$ is $x = 0.3 y - 12$ .

(a)
One of the students reads for 8 hours each week. Use an appropriate regression line to estimate this student's score in the test.
2 marks
(b)
Another student scored 70 in the test. Use an appropriate regression line to estimate the number of hours this student reads each week.
2 marks
(c)
State whether Pearson's product-moment correlation coefficient for these data is positive or negative. Give a reason for your answer.
1 mark

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Question 2 · AA-00397Paper 2 · 5 marks

The following table shows the distance from the coast, $x$ km, and the annual rainfall, $y$ mm, at eight weather stations in a region.

Distance from coast ( $x$ km)	5	18	32	47	60	78	95	120
Annual rainfall ( $y$ mm)	1420	1310	1180	1120	980	890	760	640

(a)
The regression line of $x$ on $y$ has equation $x = c y + d$ . Find the value of $c$ and the value of $d$ .
2 marks
(b)
A new weather station in the region records an annual rainfall of 1050 mm. Use your answer to part (a) to estimate the distance of this station from the coast.
2 marks
(c)
Write down the value of $r$ , Pearson's product-moment correlation coefficient, for these data.
1 mark

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Question 3 · AA-00396Paper 1 · 6 marks

The members of a running club record their resting heart rate, $h$ beats per minute, and the time, $t$ minutes, that they take to run 1 km.

For these data, the regression line of $h$ on $t$ is $h = 4 t + 40$ and the regression line of $t$ on $h$ is $t = 0.15 h - 4$ .

(a)
Find the mean resting heart rate and the mean time to run 1 km for the members of the club.
4 marks
(b)
One member of the club has a resting heart rate of 52 beats per minute. Use an appropriate regression line to estimate the time this member takes to run 1 km.
2 marks

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Question 4 · AA-00398Paper 2 · 6 marks

At a river gauging station, a hydrologist records the depth of the water, $x$ metres, and the speed of the current, $y$ m s $^{- 1}$ , on nine days. The results are shown in the following table.

Depth ( $x$ m)	0.8	1.1	1.3	1.6	1.9	2.2	2.4	2.8	3.1
Speed ( $y$ m s $^{- 1}$ )	0.62	0.95	0.78	1.20	0.98	1.42	1.18	1.55	1.48

(a)
The regression line of $y$ on $x$ has equation $y = a x + b$ . Find the value of $a$ and the value of $b$ .
2 marks
(b)
On another day, the depth gauge is broken. The speed of the current is 1.5 m s $^{- 1}$ . The hydrologist estimates the depth by solving $1.5 = a x + b$ for $x$ , which gives $x = 2.92$ m.
Explain why this method is not appropriate.
1 mark
(c)
Use an appropriate regression line to estimate the depth of the water on this day.
3 marks

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Question 5 · AA-00400Paper 1 · 8 marks

For a set of bivariate data, the two regression lines have equations

L_{1} : x - 2 y + 8 = 0 and L_{2} : 3 x - 2 y - 4 = 0.

One of these lines is the regression line of $y$ on $x$ , and the other is the regression line of $x$ on $y$ .

(a)
Find the mean point $(\overset{ˉ}{x}, \overset{ˉ}{y})$ of the data.
3 marks
(b)
It is given that if $y = a x + b$ is the regression line of $y$ on $x$ and $x = c y + d$ is the regression line of $x$ on $y$ , then $a c = r^{2}$ , where $r$ is Pearson's product-moment correlation coefficient.
Determine which of $L_{1}$ and $L_{2}$ is the regression line of $y$ on $x$ . Justify your answer.
3 marks
(c)
Find the exact value of $r$ .
2 marks

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Question 6 · AA-00399Paper 2 · 7 marks

On a busy road, the number of vehicles passing a monitoring station per hour can be modelled by

V (t) = 1350 - 950 \cos (\frac{π (t - 5)}{12}), 0 \leq t < 24,

where $t$ is the number of hours after midnight.

The station also measures the concentration of nitrogen dioxide in the air, $x$ μg m $^{- 3}$ . The following table shows the number of vehicles per hour, $y$ , and the concentration of nitrogen dioxide at eight different times.

Vehicles per hour ( $y$ )	420	760	880	1150	1390	1620	1880	2080
Nitrogen dioxide ( $x$ μg m $^{- 3}$ )	22	31	35	44	52	58	66	71

(a)
Find an appropriate regression equation that would allow the station to estimate the concentration of nitrogen dioxide when the number of vehicles per hour is known.
3 marks
(b)
Find the maximum number of vehicles per hour according to the model, and the value of $t$ at which it occurs.
2 marks
(c)
Hence estimate the concentration of nitrogen dioxide at the time found in part (b).
1 mark
(d)
Comment on the reliability of your estimate in part (c).
1 mark

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