Test the claim that the proportion of men who own cats is larger than \( 10 \% \) at the .05 significance level. The null and alternative hypothesis would be: \( \begin{array}{l}H_{0}: p=0.1 \quad H_{0}: \mu=0.1 \quad H_{0}: p=0.1 \quad H_{0}: \mu=0.1 \quad H_{0}: \mu=0.1 \quad H_{0}: p=0.1 \\ H_{1}: p<0.1 \quad H_{1}: \mu \neq 0.1 \quad H_{1}: p>0.1 \quad H_{1}: \mu>0.1 \quad H_{1}: \mu<0.1 \quad H_{1}: p \neq 0.1 \\ \text { The test is: } \\ \text { right-tailed left-tailed two-tailed }\end{array} \)

To test the claim that the proportion of men who own cats is **larger** than \( 10\% \) at the \( \alpha = 0.05 \) significance level, we need to establish the appropriate hypotheses and determine the type of test to perform. ### **Hypotheses** 1. **Null Hypothesis (\( H_0 \))**: This represents the statement of no effect or status quo. In this case, it asserts that the proportion of men who own cats is equal to \( 10\% \). \[ H_0: p = 0.1 \] 2. **Alternative Hypothesis (\( H_1 \))**: This represents the claim we want to test, which states that the proportion is **greater** than \( 10\% \). \[ H_1: p > 0.1 \] ### **Type of Test** Since the alternative hypothesis is testing whether the proportion is **greater** than a specific value, this is a **right-tailed** test. ### **Summary** - **Null Hypothesis (\( H_0 \))**: \( p = 0.1 \) - **Alternative Hypothesis (\( H_1 \))**: \( p > 0.1 \) - **Type of Test**: Right-tailed ### **Steps for the Test** 1. **Calculate the Test Statistic**: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] where: - \( \hat{p} \) = Sample proportion - \( p_0 \) = Hypothesized population proportion (\( 0.1 \)) - \( n \) = Sample size 2. **Determine the Critical Value**: - For a right-tailed test at \( \alpha = 0.05 \), the critical \( z \)-value is approximately \( 1.645 \). 3. **Make a Decision**: - If \( z \) > \( 1.645 \), reject \( H_0 \). - Otherwise, do not reject \( H_0 \). 4. **Conclusion**: - Based on the test statistic and critical value, conclude whether there is sufficient evidence to support the claim that the proportion of men who own cats is greater than \( 10\% \). ### **Example** Suppose you surveyed \( 200 \) men, and \( 30 \) of them own cats. - Sample proportion: \[ \hat{p} = \frac{30}{200} = 0.15 \] - Test statistic: \[ z = \frac{0.15 - 0.1}{\sqrt{\frac{0.1 \times 0.9}{200}}} \approx \frac{0.05}{0.0212} \approx 2.36 \] - Since \( 2.36 > 1.645 \), we reject \( H_0 \). - **Conclusion**: There is sufficient evidence at the \( 0.05 \) significance level to support the claim that the proportion of men who own cats is greater than \( 10\% \).