Suppose the confidence interval of θ\theta takes the form [A,B][A,B], if

P(A≤θ≤B)=1−α\mathbb{P}(A\le \theta\le B)=1-\alpha

then 1−α1-\alpha is called the confidence level of CI, i.e. the CI covers θ\theta with 1−α1-\alpha probability.

Margin of Error

Since θ\theta can be larger or smaller than a point estimator θ^\hat\theta, CI typically use the form

θ^±ME\hat\theta \plusmn ME

  • the width of CI w=2×MEw=2\times ME
  • the upper confidence limit UCL=θ^+MEUCL=\hat\theta + ME
  • the lower confidence limit LCL=θ^−MELCL=\hat\theta - ME

CI construction is equivalent to collect all parameter values not rejected by the hypothesis test, which means that μ0∉CI  ⟹  reject H0:μ=μ0\mu_0 \not\in \text{CI} \implies \text{reject }H_0:\mu=\mu_0. CI gives decision for all possible hypothesis at once.

Next, we analyze the CI in several settings:

Finite Population Correction

If n>0.05Nn\gt 0.05N, we adjust the variance

σ^xˉ2=s2nN−nn\hat{\sigma}^2_{\bar{x}}=\frac{s^2}{n}\frac{N-n}{n}

and the 1−α1-\alpha CI is

xˉ±tn−1,α/2σ^xˉ\bar{x} \plusmn t_{n-1,\alpha/2}\hat{\sigma}_{\bar{x}}

For “one proportion, large samples”:

p^±zα/2σ^p^\hat{p}\plusmn z_{\alpha/2}\hat{\sigma}_{\hat{p}}

where

σ^p^2=p^(1−p^)n−1N−nN−1\hat{\sigma}_{\hat{p}}^2 = \frac{\hat{p}(1-\hat{p})}{n-1}\frac{N-n}{N-1}

is an unbiased estimator of Var(p^)Var(\hat{p}).

Sample Size Determination

For One normal mean, known variance case, to make a (1−α)(1-\alpha) CI for μ\mu extend a distance ME\text{ME} on each side of xˉ\bar{x}, we need

n=⌈zα/22σ2ME2⌉n=\lceil \frac{z_{\alpha/2}^2\sigma^2}{\text{ME}^2} \rceil

samples.

For One proportion, Large Samples case, a sample size of

n=0.252zα/22ME2n=\frac{0.25^2 z_{\alpha/2}^2}{\text{ME}^2}

can guarantee that CI extends no more than ME\text{ME} on each side of p^\hat p

ME\text{ME} reported in the media includes only the sampling error in p^\hat p and does not include any errors due to biased or inadequate samples.