The normal distribution as a model for continuous data, finding probabilities, the standard normal distribution, using the inverse normal, and approximating the binomial.

What this dot point is asking

Edexcel wants you to use the normal distribution $N(\mu, \sigma^2)$ as a model for continuous data, find probabilities with a calculator, standardise to the standard normal $Z \sim N(0, 1)$, use the inverse normal to find values or unknown $\mu$ or $\sigma$, recognise when a normal model is suitable, and use the normal approximation to the binomial.

The answer

The normal model

Standardising

Inverse normal and approximation

The inverse normal finds the value of $X$ for a given cumulative probability, which lets you solve for an unknown $\mu$ or $\sigma$ from a known proportion. For a binomial with large $n$, you can approximate $B(n, p)$ by $N(np, np(1 - p))$, applying a continuity correction.

When a normal model is suitable

A normal model fits data that are roughly symmetric and bell-shaped about a central value, with most values near the mean and few in the tails. Heights, masses and measurement errors often qualify. It is a poor model for strongly skewed data, for counts that cannot be negative but have a long upper tail, or for data with two separate peaks. The normal approximation to the binomial works best when $n$ is large and $p$ is close to $0.5$, so that $np$ and $n(1 - p)$ are both reasonably large.

Examples in context

Try this

Q1. For $X \sim N(50, 16)$, find $P(X > 54)$. [3 marks]

• Cue. $Z = \dfrac{54 - 50}{4} = 1$, so $P(Z > 1) \approx 0.159$.

Q2. State the approximating normal distribution for $X \sim B(100, 0.5)$. [2 marks]

• Cue. $np = 50$, $np(1 - p) = 25$, so $N(50, 25)$.