4.9 The normal distribution questions for IB Maths AA

52 questions on The normal distribution, 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	HL only	All
Paper 1	11	1	12
Paper 2	38	2	40
Every paper	49	3	52

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-00429Paper 1 · 5 marks

Amira and Ben sat end-of-year tests in different subjects. The marks in each test were normally distributed.

In history, the mean mark was $62$ and the standard deviation was $8$ . Amira scored $74$ .
In chemistry, the mean mark was $70$ and the standard deviation was $5$ . Ben scored $81$ .

(a)(i)
Find the standardized value ( $z$ -value) of Amira's mark.
1 mark
(a)(ii)
Find the standardized value of Ben's mark.
1 mark
(b)
State which student performed better relative to the other students who sat the same test. Give a reason for your answer.
1 mark
(c)
Carlos also sat the history test. His mark had a standardized value of $- 0.75$ . Find Carlos's mark.
2 marks

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Question 2 · AA-00004Paper 2 · 5 marks

The masses of apples from an orchard are normally distributed with mean 180 g and standard deviation 12 g.

(a)
Find the probability that a randomly chosen apple has a mass of more than 200 g.
2 marks
(b)
Apples heavier than $k$ grams are sold as “premium”. 15% of apples are premium. Find $k$ .
3 marks

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Question 3 · AA-00433Paper 1 · 8 marks

Let $Z \sim N (0, 1)$ . It is given that $P (Z > 1.5) = 0.0668$ and $P (Z < - 0.5) = 0.3085$ .

Let $X \sim N (μ, σ^{2})$ . It is known that $P (X > 73) = 0.0668$ and $P (X < 57) = 0.3085$ .

(a)
Find the value of $μ$ and the value of $σ$ .
4 marks
(b)
Find $P (57 < X < 73)$ .
2 marks
(c)
Find $P (X > 49)$ .
2 marks

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Question 4 · AA-00405Paper 2 · 16 marks

A farmer records the total rainfall, $x$ mm, during the growing season and the wheat yield, $y$ tonnes per hectare, on her farm for ten years. The results are shown in the following table.

Rainfall ( $x$ mm)	412	468	495	530	548	575	602	640	688	725
Yield ( $y$ t ha $^{- 1}$ )	2.9	3.4	3.3	3.9	3.6	4.2	4.0	4.6	4.5	5.1

(a)
Find the value of Pearson's product-moment correlation coefficient, $r$ .
2 marks
(b)(i)
Find the equation of the regression line of $y$ on $x$ .
2 marks
(b)(ii)
Find the equation of the regression line of $x$ on $y$ .
2 marks
(c)
In the following year, the rainfall during the growing season was 520 mm. Use an appropriate regression line to estimate the wheat yield in that year.
2 marks
(d)
The rainfall records for one earlier year were lost. The wheat yield in that year was 4.2 t ha $^{- 1}$ . Use an appropriate regression line to estimate the rainfall during the growing season in that year.
2 marks
(e)
The rainfall during the growing season, $R$ mm, in this region is normally distributed with mean 570 mm and standard deviation 90 mm.
Find the probability that the rainfall in a randomly chosen growing season is less than 450 mm.
2 marks
(f)
In 90% of growing seasons, the rainfall is more than $k$ mm. Find the value of $k$ .
2 marks
(g)
Use an appropriate regression line to estimate the wheat yield in a growing season with $k$ mm of rainfall.
2 marks

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Question 5 · AA-00434Paper 1 · 5 marks

Let $X \sim N (20, 4^{2})$ and $Y \sim N (35, σ^{2})$ , where $σ > 0$ .

(a)
Find the standardized value of $x = 26$ .
1 mark
(b)
It is given that $P (Y > 44) = P (X > 26)$ . Find the value of $σ$ .
2 marks
(c)
Find the value of $c$ such that $P (Y < c) = P (X > 26)$ .
2 marks

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Question 6 · AA-00414Paper 2 · 8 marks

The lengths, $X$ cm, of cucumbers grown on a farm are normally distributed with mean 28 cm and standard deviation 2.5 cm.

(a)
Find the probability that a randomly selected cucumber is shorter than 30 cm.
2 marks
(b)
Given that a randomly selected cucumber is shorter than 30 cm, find the probability that it is longer than 26 cm.
3 marks
(c)
Only cucumbers longer than 25 cm are sold to a supermarket. Find the value of $k$ such that half of the cucumbers sold to the supermarket are longer than $k$ cm.
3 marks

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