## Hypothesis Test of Proportion

Here we want to compare the population proportion with a particular value . Then our hypothesis testing problem becomes

For example, in order for vaccine to be approved for widespread use, it must be established that the probability of serious adverse reaction must be less than . In this case we should set `` versus '' to see whether we can reject in favor of .

Let be the frequency of the specified type in categorical data of size . In the above vaccine example will be the number of participants who suffered adverse reaction among participants. Then the test statistic is calculated as

The appropriateness of the test is established by adequately large sample size , ensured by and . Provided that the null hypothesis is true, the test statistic has approximately the standard normal distribution. We can reject with significance level if the test statistic falls into the critical region indicated in the table below.

Hypotheses | Critical region to reject |

versus . | |

versus . | |

versus . |

Alternatively we can proceed to construct the p-value = , and reject when the p-value is less than of your choice.

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

Then the two different formulas

are available for the -level confidence interval. Although the first formula is easier to calculate, the second is known to be more accurate and widely used.