Summary of Exact Binomial Inference (Sampling from a Binomial Process).
Let X represent the number of successes in the sample and \(\pi\) the probability of success for a binomial random process.
To test \(H_0: \pi = \pi_0\)
We can calculate a p-value based on the binomial distribution with parameters n and \(\pi_0\text{.}\) The p-value can be one-sided or two-sided based on the statement of the research conjecture.
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If \(H_a: \pi > \pi_0\text{:}\) p-value = P(X β₯ observed)
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If \(H_a: \pi < \pi_0\text{:}\) p-value = P(X β€ observed)
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If \(H_a: \pi \neq \pi_0\text{:}\) p-value = sum of both tail probabilities using a method like "small p-values"
(100 Γ C)% Confidence Interval for \(\pi\)
The set of values such that the two-sided p-value based on the observed count is larger than the \((1 - C)\) cut-off.
Technology
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One Proportion Inference applet for approximate and exact binomial probability (p-value)
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R, ISCAM Workspace:
iscambinomtest(observed, n, hypothesized=Ο_0, alternative="greater", "less," or "two.sided", conf.level)Can enter either sample count or sample proportion for "observed." If you donβt specify a hypothesized value and alternative, be sure to label the confidence level. -
JMP:For a one-sided p-value: Analyze > Distribution (raw or tallied data using Freq)For a confidence interval: ISCAM Journal file > Confidence Interval for One Proportion using Summary Stats: specify the number of successes and the sample size
