Step 1 :The problem is asking to test the manufacturer's claim that more than 91% of patients taking the drug are healed within 8 weeks. The null hypothesis is that the proportion of patients healed is equal to 91% and the alternative hypothesis is that the proportion of patients healed is greater than 91%.
Step 2 :To determine the test statistic, we need to use the formula for the z-score which is \((\hat{p} - p_0) / \sqrt{(p_0 * (1 - p_0)) / n}\), where \(\hat{p}\) is the sample proportion, \(p_0\) is the population proportion under the null hypothesis, and \(n\) is the sample size.
Step 3 :In this case, \(\hat{p}\) is 221/239, \(p_0\) is 0.91, and \(n\) is 239.
Step 4 :Let's calculate the test statistic: \(\hat{p} = 0.9246861924686193\), \(p_0 = 0.91\), \(n = 239\), \(z_0 = 0.7933525374353512\).
Step 5 :The test statistic, \(z_0\), is approximately 0.79 when rounded to two decimal places. This value will be used to determine the p-value and make a decision about the null hypothesis.
Step 6 :Final Answer: \(z_{0}=\boxed{0.79}\)