Construct and interpret confidence intervals for a population mean and find the required sample size

What this dot point is asking

SCSA wants you to build the interval from a sample, choose the right critical value, interpret the level correctly, and rearrange to find the sample size needed for a target margin of error.

The interval

where $z$ is the standard-normal critical value for the chosen level: $z \approx 1.645$ for $90\%$, $z \approx 1.96$ for $95\%$, and $z \approx 2.576$ for $99\%$. The half-width $z\tfrac{\sigma}{\sqrt{n}}$ is the margin of error $E$.

Interpreting the confidence level

Confidence versus precision

A higher confidence level uses a larger $z$, widening the interval: more confidence costs precision. A larger sample reduces the standard error, narrowing the interval. The two competing effects are both visible in the margin of error $z\tfrac{\sigma}{\sqrt{n}}$.

Finding the required sample size

Set the margin of error to the target $E$ and solve for $n$:

Round $n$ up to the next whole number, since a fractional sample is not possible and rounding down would exceed the target margin.