Use the normal distribution and the 68-95-99.7 rule, standardise to z-scores, and construct and interpret sample proportions and confidence intervals.

What this dot point is asking

You must apply the empirical rule, calculate and use z-scores, find sample proportions, and construct and explain confidence intervals.

The normal distribution

A normal distribution is the symmetric, bell-shaped curve described by its mean $\mu$ and standard deviation $\sigma$. It is symmetric about $\mu$, where it also peaks.

Standardising with z-scores

A z-score says how many standard deviations a value is above ($z > 0$) or below ($z < 0$) the mean. It lets you compare values from different normal distributions.

For example, a score of $86$ in a test with $\mu = 70, \sigma = 8$ gives $z = (86 - 70)/8 = 2$, so it is $2$ standard deviations above the mean, a strong result.

Sample proportions

The sample proportion is $\hat{p} = \dfrac{x}{n}$, where $x$ successes occur in a sample of size $n$. It is a point estimate of the unknown population proportion $p$. Larger samples give estimates that vary less from sample to sample.

Confidence intervals for a proportion

A confidence interval is a range, built from one sample, that we are a stated percentage confident contains the true population proportion.