Questions: A drug company claims that an allergy medication causes headaches in 5% of those who take it. A medical researcher believes that more than 5% of those who take the drug actually get headaches. a. Identify the population.

Solution Steps

We are testing the claim made by a medical researcher that the proportion of individuals who experience headaches after taking the allergy medication is greater than \(5\%\).

• Null Hypothesis (\(H_0\)): \(p = 0.05\)
• Alternative Hypothesis (\(H_a\)): \(p > 0.05\)

The test statistic \(Z\) is calculated using the formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Substituting the values:

• \(\hat{p} = 0.06\) (sample proportion)
• \(p_0 = 0.05\) (hypothesized population proportion)
• \(n = 100\) (sample size)

We find:

\[ Z = \frac{0.06 - 0.05}{\sqrt{\frac{0.05(1 - 0.05)}{100}}} = 0.4588 \]

The P-value associated with the test statistic \(Z = 0.4588\) is calculated to be:

\[ \text{P-value} = 0.3232 \]

For a significance level of \(\alpha = 0.05\) in a one-tailed test, the critical value is:

\[ Z_{critical} = 1.6449 \]

The critical region is defined as:

\[ Z > 1.6449 \]

Since the calculated test statistic \(Z = 0.4588\) is less than the critical value \(1.6449\), we fail to reject the null hypothesis. Additionally, the P-value \(0.3232\) is greater than \(\alpha = 0.05\).

Final Answer