Question

a certain drug is used to treat asthma. in a clinical trial of the drug, 21 of 270 treated subjects experienced headaches (based on data from the manufacturer). the accompanying calculator display shows results from a test of the claim that less than 12% of treated subjects experienced headaches. use the normal distribution as an approximation to the binomial distribution and assume a 0.05 significance level to complete parts (a) through (e) below. 1 - propztext prop < 0.12 z = - 2.134966231 p = 0.0163818843 p ^ = 0.0777777778 n = 270 left - tailed test b. what is the test statistic? z = - 2.13 (round to two decimal places as needed.) c. what is the p - value? p - value = 0.016 (round to three decimal places as needed.) d. what is the null hypothesis, and what do you conclude about it? identify the null hypothesis. a. h₀: p > 0.12 b. h₀: p < 0.12 c. h₀: p = 0.12 d. h₀: p ≠ 0.12 decide whether to reject the null hypothesis. choose the correct answer below. a. fail to reject the null hypothesis because the p - value is less than or equal to the significance level, α b. reject the null hypothesis because the p - value is less than or equal to the significance level, α c. reject the null hypothesis because the p - value is greater than the significance level, α d. fail to reject the null hypothesis because the p - value is greater than the significance level, α

Explanation:

Step1: Identify the null hypothesis

The null hypothesis \(H_0\) for a proportion test is a statement of equality. When testing a claim about a proportion \(p\), if the claim is \(p < 0.12\), the null hypothesis is \(H_0:p = 0.12\).

Step2: Determine the decision rule for the null hypothesis

The significance level \(\alpha=0.05\). The decision rule for a hypothesis test is: if \(P - value\leq\alpha\), reject \(H_0\); if \(P - value>\alpha\), fail to reject \(H_0\). Here, \(P - value = 0.016\) and \(\alpha = 0.05\). Since \(0.016<0.05\) (i.e., \(P - value\leq\alpha\)).

Answer:

• For the null hypothesis: C. \(H_0:p = 0.12\)
• For the decision about the null hypothesis: B. Reject the null hypothesis because the P - value is less than or equal to the significance level, \(\alpha\)