e-Statistics

Testing a Proportion

Here we compare the population proportion p of specific type with particular value . For example, in order for vaccine to be approved for widespread use, it must be established that the probability p of serious adverse reaction must be less than a particular value . Then we should set “ $H_0:\: p = p_0$ versus $H_A:\: p < p_0$ ” and see whether we can reject in favor of .

In general, our hypothesis test problem becomes

$H_A:\hspace{0.05in}p$

Count data consists of the frequency X = of the specified type, and the size n = of data. In the above vaccine example X will be the number of participants who suffered adverse reaction among

participants.

The appropriateness of z-test is assumed for a adequately large sample size n, ensured by $np_0 \ge 5$ and $n(1-p_0) \ge 5$ . Provided that the null hypothesis is true, the test statistic

$Z = \dfrac{X - np_0}{\sqrt{np_0(1-p_0)}} =$ (with continuity correction)

has approximately the standard normal distribution, which allows us to calculate p-value = by the standard normal distribution.

When is rejected, we want to further investigate the confidence interval for the population proportion p. We have the point estimate of p by

$\hat{p} = \dfrac{X}{n} =$

The $(1-\alpha)$ -level confidence interval

$\displaystyle \hat{p} \pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} =$ ( , )

can be calculated for the population proportion.

Consider the frequency function f(k) from binomial distribution with parameter $(n,p_0)$ . Provided that the null hypothesis is true, $X=k$ can be viewed as an extreme event if the sum of f(i) with $f(i)\le f(k)$ is small. Thus, we can proceed to calculate it as the p-value = , and reject when the p-value is less than $\alpha$ of your choice.