Subsection 5.6 Choice of Procedures for Analyzing One Proportion
| Aspect | Random Process | Finite Population |
|---|---|---|
| Study design | One binary variable | |
| Parameter | \(\pi\) = probability of success | \(\pi\) = population proportion |
| Null Hypothesis | \(H_0: \pi = \pi_0\) | |
| Simulation | Random sample from binomial process | Random sample from a finite population |
| Exact distribution | Binomial distribution (\(n\text{,}\) \(\pi_0\)): \(E(\hat{p}) = \pi\text{,}\) \(SD(\hat{p}) = \sqrt{\pi(1-\pi)/n}\) | Hypergeometric (\(N\text{,}\) \(M\text{,}\) \(n\)): \(E(\hat{p}) = M/N\text{,}\) \(SD(\hat{p}) = \sqrt{\pi(1-\pi)/n}\times\)\(\sqrt{(N-n)/(N-1)}\) |
| Valid when (for z procedures) | At least 10 successes and 10 failures | Population size > 20n; At least 10 successes and 10 failures |
| Standardized statistic | \(z_0 = (\hat{p} - \pi_0)/\sqrt{\pi_0(1-\pi_0)/n}\) | |
| Confidence interval | Exact Binomial: all plausible values with two-sided p-value > 0.05; Wald: \(\hat{p} \pm z^*\sqrt{\hat{p}(1-\hat{p})/n}\text{;}\) Adjusted Wald: \(\tilde{p} \pm z^*\sqrt{\tilde{p}(1-\tilde{p})/\tilde{n}}\) where \(\tilde{p} = (X + 0.5z^{*2})/(n + z^{*2})\) and \(\tilde{n} = n + z^{*2}\text{.}\) For 95% confidence, add two successes and two failures. |
|
| R Commands | iscamonepropztest • Observed (either the number of successes or sample proportion) • n (sample size) • hypothesized probability (\(\pi_0\)) • alternative ("less", "greater", or "two.sided") • Optional: conf.level(s) |
iscamhyperprob • k (observed number of successes) • total (population size) • succ (hypothesized number of successes in population) • n (sample size) • lower.tail (TRUE or FALSE) |
| JMP | Analyze > Distribution (one-sided p-values) | Formula > Discrete Probability > Hypergeometric Distribution |
| TBI Applet | One proportion | |
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