A confidence interval (CI) is a range of values, calculated from sample data, that is likely to contain the true population parameter. The confidence level (usually 95%) expresses how often such intervals capture the true value if the process were repeated many times.
For a 95% CI, \(z \approx 1.96\) (from the standard normal distribution).
Margin of error: \(E = 1.96 \cdot \dfrac{\sigma}{\sqrt{n}}\)
where \(\hat{p}\) is the sample proportion and \(n\) is the sample size.
A random sample of 64 daily temperature readings has \(\bar{x} = 18.5°\text{C}\) and population \(\sigma = 3.2°\text{C}\).
We are 95% confident the true mean temperature lies between 17.7°C and 19.3°C.
A survey of 200 voters finds 112 support a policy. \(\hat{p} = 112/200 = 0.56\).
We are 95% confident the true proportion of supporters is between 49.1% and 62.9%.
Larger \(n\) \(\Rightarrow\) smaller \(SE\) \(\Rightarrow\) narrower CI \(\Rightarrow\) more precise estimate.
To halve the margin of error, you need four times the sample size (since \(SE \propto 1/\sqrt{n}\)).
REMEMBER: “95% confident” means: if this sampling procedure were repeated many times, 95% of such intervals would contain the true population parameter.
EXAM TIP: Show all four steps: (1) calculate \(\hat{p}\) or state \(\bar{x}\), (2) calculate \(SE\), (3) compute margin of error, (4) state the interval and interpret it in context.
