A certain drug can be used to reduce the acid produced by the body and heal damage to the esophagus due to acid reflux. The manufacturer of the drug claims that more than $91 \%$ of patients taking the drug are healed within 8 weeks. In clinical trials, 221 of 239 patients suffering from acid reflux disease were healed after 8 weeks. Test the manufacturer's claim at the $\alpha=0.05$ level of significance,
Click here to view the standard normal distribution table (page 1). Click here to view the standard normal distribution table (page 2).
Because $n p_{0}\left(1-p_{0}\right)=19.6> 10$, the sample size is less than ${ }^{\top} 5 \%$ of the population size, and the sample can be reasonably assumed to be random, $\$$ the requirements for testing the hypothesis are satisfied.
(Round to one decimal place as needed)
What are the null and alternative hypotheses?
$\mathrm{H}_{0}: \mathrm{p}=.91$ versus $\mathrm{H}_{1}: \mathrm{p}> 91$
(Type integers or decimals. Do not round.)
Determine the test statistic, $z_{0}$.
$z_{0}=\square$ (Round to two decimal places as needed.)
2
Answer
Final Answer: \(z_{0}=\boxed{0.79}\)
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: \(z_{0}=\boxed{0.79}\)
