Confidence intervals for the difference between two proportions

The statistical analysis approach is to calculate 95% confidence intervals for the proportion of participants in each group (placebo and combined active) reporting a headache. This analysis approach is reasonable because the sample size is sufficiently large (that is, the values, pn, in each group are at least five). Satisfying this assumption enables us to use the Z distribution for the reliability factor. 

The first step is to calculate the point estimate of the proportion. For the placebo group the proportion is 0.06. The second step is to calculate the standard error. For this estimator the standard error is calculated as follows  

The third component of the interval estimate is the reliability factor. As we are calculating a two-sided 95% confidence interval, we select the value of Z from Table 8.3 corresponding to a of 0.05, that is, 1.96. 

With all of the components now available, the last step is to calculate the confidence interval. The lower limit is 0.06 - 1.96(0.02) = 0.02. The upper limit is 0.06 + 1.96(0.02) = 0.10. We write the 95% confidence interval as (0.02, 0.10). Repeating these steps for the combined active dose group, we obtain a 95% confidence interval of (0.04, 0.12). (We leave it to you to verify this calculation.) 

Using these two confidence intervals we can now make some conclusions about the unknown population proportion of participants who experience headache after exposure in each group. In the case of the placebo group, we are 95% confident that the population proportion of participants experiencing a headache is enclosed in the interval (0.02, 0.10). For the combined active dose group, we are 95% confident that the population proportion of participants experiencing a headache is enclosed in the interval (0.04, 0.12). Although it may initially have seemed that there may be an increased risk of headache associated with the active treatment. 

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