Model continuous data with the normal distribution, standardise to the Z-distribution to find probabilities, and find values from given probabilities

What this dot point is asking

SEAB wants you to model continuous data with the normal distribution, standardise to the standard normal $Z$-distribution to find probabilities, solve inverse problems (find a value given a probability), and combine independent normal variables.

The answer

The normal distribution

A continuous variable $X \sim \mathrm{N}(\mu, \sigma^2)$ has a symmetric bell-shaped curve centred at the mean $\mu$ with spread set by the standard deviation $\sigma$. Probabilities are areas under the curve, so $\mathrm{P}(X = a) = 0$ for any single point and only intervals carry probability.

Standardising to Z

Any normal variable converts to the standard normal $Z \sim \mathrm{N}(0, 1)$ by

A $z$-score measures how many standard deviations a value lies from the mean. Then $\mathrm{P}(X < a) = \mathrm{P}\left(Z < \dfrac{a - \mu}{\sigma}\right)$, read from the standard normal (or the graphing calculator).

Finding probabilities

Sketch the bell curve, shade the required region, and express it using the cumulative function. Use symmetry ($\mathrm{P}(Z < -z) = \mathrm{P}(Z > z)$) and the complement for tails. The graphing calculator gives normal probabilities directly.

Inverse problems

To find a value $a$ such that $\mathrm{P}(X < a) = p$, find the $z$-value with that cumulative probability, then unstandardise: $a = \mu + z\sigma$.

Combining normal variables

If $X \sim \mathrm{N}(\mu_X, \sigma_X^2)$ and $Y \sim \mathrm{N}(\mu_Y, \sigma_Y^2)$ are independent, then $aX + bY$ is normal with mean $a\mu_X + b\mu_Y$ and variance $a^2\sigma_X^2 + b^2\sigma_Y^2$ (variances add, with squared coefficients).

Examples in context

Example 1. Setting a pass mark. An examiner choosing a mark so that the top $15\%$ achieve a distinction solves an inverse normal problem, finding the $z$ for the $85$th percentile and unstandardising to the raw mark.

Example 2. Tolerance in manufacturing. A component dimension $\mathrm{N}(\mu, \sigma^2)$ is acceptable within a tolerance band; standardising the limits gives the proportion within specification, the everyday quality-control calculation.

Try this

Q1. For $X \sim \mathrm{N}(100, 15^2)$, find the $z$-score of $X = 130$. [1 mark]

• Cue. $z = \dfrac{130 - 100}{15} = 2$.

Q2. For $Z \sim \mathrm{N}(0, 1)$, find $\mathrm{P}(Z > 1)$. [2 marks]

• Cue. $1 - 0.8413 = 0.1587$.

Q3. Independent variables $X \sim \mathrm{N}(5, 4)$ and $Y \sim \mathrm{N}(3, 9)$. State the distribution of $X + Y$. [2 marks]

• Cue. $\mathrm{N}(8, 13)$ (means and variances add).