State uncertainty honestly

When the population standard deviation is known and the sampling model is normal or adequately approximated by normality, a confidence interval for the mean combines the sample mean with a critical value times its standard error.

A 95% confidence procedure covers the fixed population mean in about 95% of repeated independent samples. Increasing sample size narrows intervals; increasing confidence widens them. Sampling bias is not repaired by a narrow interval.

\displaystyle \overline x\pm z^*\frac\sigma{\sqrt n}

WORKED EXAMPLE

Use x̄ = 40, known σ = 10, n = 100 and z* = 1.96 to estimate the mean.

The standard error is $\frac{10}{10}$ = 1.
The margin of error is 1.96.
The interval is (38.04, 41.96).

Assumed knowledge

Use the sampling distribution of a mean and its standard error. Find probabilities and critical values for a standard normal variable, and solve inequalities involving square roots and sample size.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

State uncertainty honestly

Use x̄ = 40, known σ = 10, n = 100 and z* = 1.96 to estimate the mean.

Put the idea to work.