Solving Normal Distribution Problems — 9709 Statistics 1 (5.5) — Mathematics

Introduction

The Normal distribution is one of the most frequently examined topics in 9709 Statistics 1. Once you understand its shape and symmetry, the key skill is translating between the original variable $X$ and the standard Normal variable $Z$ — a process called standardisation. This note covers both directions of problem:

Given $x_1$ , $\mu$ , $\sigma$ → find a probability such as $P(X > x_1)$ .
Given a probability → find a relationship between $x_1$ , $\mu$ , and $\sigma$ , or determine the unknown parameter(s).

CAIE examiners explicitly require full standardisation working to be shown; omitting the $Z = \dfrac{X - \mu}{\sigma}$ step will cost marks even if the final answer is correct.

Core Concept

If $X \sim N(\mu, \sigma^2)$ , the standardisation formula converts any value $x_1$ of $X$ into a value $z_1$ of the standard Normal variable $Z \sim N(0,1)$ :

Z = \frac{X - \mu}{\sigma}

All probability lookups are then made using the $\Phi(z)$ table, which gives $P(Z \leq z)$ for $z \geq 0$ .

Key symmetry results (essential for the exam):

Probability form	Rewritten using $\Phi$
$P(Z \leq z)$	$\Phi(z)$
$P(Z \geq z)$ for $z > 0$	$1 - \Phi(z)$
$P(Z \leq -z)$ for $z > 0$	$1 - \Phi(z)$
$P(Z \geq -z)$ for $z > 0$	$\Phi(z)$
$P(z_1 \leq Z \leq z_2)$	$\Phi(z_2) - \Phi(z_1)$

Finding unknown parameters: When a probability is given, you reverse-engineer $z$ from the tables (using the percentage-point column or inverse lookup), then set up an equation using $z = \dfrac{x_1 - \mu}{\sigma}$ and solve for the unknown.

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Introduction

Core Concept

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