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}: p0.1 \quad H_{1}: \mu0.1 \quad H_{1}: \mu

Question

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: $$ \begin{array}{l}H_{0}: p=0.1 \quad H_{0}: \mu=0.1 \quad H_{0}: p=0.1 \quad H_{0}: \mu=0.1 \quad H_{0}: \mu=0.1 \quad H_{0}: p=0.1 \ H_{1}: p<0.1 \quad H_{1}: \mu eq 0.1 \quad H_{1}: p>0.1 \quad H_{1}: \mu>0.1 \quad H_{1}: \mu<0.1 \quad H_{1}: p eq 0.1 \ ext { The test is: } \ ext { right-tailed left-tailed two-tailed }\end{array} $$

UpStudy AI · Accepted Answer

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 $$ 10\% $$.
**Hypotheses:** - **Null Hypothesis ($$ H_0 $$)**: The proportion of men who own cats is equal to $$ 10\% $$ ($$ p = 0.1 $$). - **Alternative Hypothesis ($$ H_1 $$)**: The proportion of men who own cats is greater than $$ 10\% $$ ($$ p > 0.1 $$). **Type of Test**: Right-tailed

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