17.3.7.2 Algorithms (One Sample Proportion Test)

Hypotheses

Let \(n\!\) be the sample size and \(n_{1}\!\)be the number of events or successes. Then the sample proportion \(\tilde{p}\!\) can be expressed:\(\tilde{p}=\frac{n_{1}}{n}\)

Let \(p\!\) be the sample proportion and the \(p_{0}\!\) is the hypothetical proportion, this function tests the hypotheses:

\(H_0:p=p_{0}\!\) vs \(H_1:p \ne p_{0}\!\), for a two-tailed test.

\(H_0: p\ge p_{0}\!\) vs \(H_1:p < p_{0}\!\), for a lower-tailed test.

\(H_0:p\le p_{0}\!\) vs \(H_1:p > p_{0}\!\), for an upper-tailed test.

Normal approximation

P Value

When \(n_{1}\ge10\!\) and \(n-n_{1}\ge10\!\), we can compute a p-value using a normal approximation of a binomial distribution. To perform the test, compute the \(z\!\) and \( p_{value}\!\) value by:

\[z=\frac{\tilde{p}-p_{0}}{\sqrt{\frac{p_{0}(1-p_{0})}{n}}}\!\]

\(p_{value}=2p(Z>|z||p=p_{0})\!\),for two tailed test

\(p_{value}=p(Z\le z|p=p_{0})\!\),for upper tailed test

\(p_{value}=p(Z\ge z|p=p_{0})\!\)for lower tailed test

Confidence Interval

confidence level is equal to \(1-\alpha\), the confidence interval for the sample proportion can be generated by:

Null Hypothesis	Confidence Interval
\[H_0:p=p_{0}\!\]	\[\left[\tilde{p}- Z_{\frac{\alpha}{2}}\sqrt{\frac{\tilde{p}(1-\tilde{p})}{n}}, \tilde{p}+ Z_{\frac{\alpha}{2}}\sqrt{\frac{\tilde{p}(1-\tilde{p})}{n}}\right]\]
\[H_0: p\ge p_{0}\!\]	\[\left[\tilde{p}- Z_{\alpha}\sqrt{\frac{\tilde{p}(1-\tilde{p})}{n}}, 1\right]\]
\[H_0:p\le p_{0}\!\]	\[\left[0, \tilde{p}+ Z_{\alpha}\sqrt{\frac{\tilde{p}(1-\tilde{p})}{n}}\right]\]

Binomial Test

Exact P_value

In Origin, the exact test for one proportion is based on the Binomial Test .

\(H_0: p\ge p_{0}\!\) \(P_{value}=p(X\le n_{1}|p_0)\)

\(H_0:p\le p_{0}\!\) \(P_{value}=p(X\ge n_{1}|p_0)\)

\(H_0:p=p_{0}\!\):

Let \(M=n*p_0\!\),

when \(n_1=M\!\) \(P_{value}=1\!\)

when \(n_1\le M\!\) \(P_{value}=P(X\le n_1)+P(X\ge n-y+1)\), where y is the count for z such that \(P(z)\le p(n_1)\) and \(n\ge z\ge \left \lfloor M \right \rfloor+1\)

when \(n_1\ge M\!\) \(P_{value}=P(X\le y-1)+P(X\ge n_1)\), where y is the count for z such that \(P(z)\le p(n_1)\) and \(0\le z\le \left \lfloor M \right \rfloor\)

Exact Confidence Interval

Exact Confidence interval: confidence levels is \(1-\alpha\)

Null Hypothesis	Confidence Interval
\[H_0:p=p_{0}\!\]	\[\left[QBETA_{(1 - \alpha/2, n_1 + 1, n - n_1)}, QBETA_{(\alpha/2, n_1, n - n_1 + 1)}\right]\]
\[H_0: p\ge p_{0}\!\]	\[\left[QBETA_{(1 - \alpha, n_1 + 1, n - n_1)}, 1\right]\]
\[H_0:p\le p_{0}\!\]	\[\left[0, QBETA_{(\alpha, n_1, n - n_1 + 1)}\right]\]

where \(QBETA\) denotes the quantile function of Beta distribution.