4.12 Standardising normal variables questions for IB Maths AA

22 questions on Standardising normal variables, 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	4	0	4
Paper 2	16	2	18
Every paper	20	2	22

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-00420Paper 2 · 15 marks

The lifetimes, $T$ hours, of a brand of rechargeable battery are normally distributed with mean 40 hours and standard deviation $σ$ hours. It is known that 10% of these batteries have a lifetime of less than 37 hours.

(a)
Show that $σ = 2.341$ , correct to four significant figures.
3 marks
(b)
Find the probability that a randomly selected battery has a lifetime of more than 44 hours.
2 marks
(c)
A battery has already lasted 42 hours. Find the probability that its lifetime is more than 45 hours.
3 marks
(d)
Find the probability that at least 7 of the batteries in a randomly selected pack have a lifetime of at least 37 hours.
3 marks
(e)
Given that at least one battery in a pack has a lifetime of less than 37 hours, find the probability that exactly two batteries in the pack have a lifetime of less than 37 hours.
4 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-00427Paper 2 · 5 marks

The time taken for a ferry to cross a lake, $T$ minutes, is normally distributed with mean $42$ minutes and standard deviation $σ$ minutes. It is found that $6 %$ of crossings take longer than $47$ minutes.

(a)
Find the value of $σ$ .
3 marks
(b)
Find the probability that a randomly chosen crossing takes less than $40$ minutes.
2 marks

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