Test the claim that the proportion of people who own cats is larger than 30% at the 0.10 significance level.

The null and alternative hypothesis would be:
H0:μ≤0.3𝐻0:𝜇≤0.3
H1:μ>0.3𝐻1:𝜇>0.3
Based on a sample of 100 people, 31% owned cats

The p-value is

Let's go through the steps to calculate the p-value for testing the given hypothesis.

### Problem Summary:
- **Claim**: The proportion of people who own cats is larger than 30%.
- **Sample size (n)**: 100
- **Sample proportion (p̂)**: 31% = 0.31
- **Significance level (α)**: 0.10
- **Null Hypothesis (H₀)**: p ≤ 0.3 (The proportion of people who own cats is 30% or less)
- **Alternative Hypothesis (H₁)**: p > 0.3 (The proportion of people who own cats is greater than 30%)

### Step 1: Calculate the Test Statistic (z)
The test statistic for a proportion is calculated using the formula:

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

Where:
- \(\hat{p}\) is the sample proportion (0.31)
- \(p_0\) is the hypothesized population proportion (0.3)
- \(n\) is the sample size (100)

Substitute the values:

\[
z = \frac{{0.31 - 0.30}}{{\sqrt{\frac{{0.3 \cdot (1 - 0.3)}}{100}}}} = \frac{{0.01}}{{\sqrt{\frac{{0.21}}{100}}}} = \frac{{0.01}}{{\sqrt{0.0021}}} = \frac{{0.01}}{{0.0458}} \approx 0.2185
\]

### Step 2: Find the P-value
The p-value is the probability of obtaining a z-value greater than 0.2185 under the standard normal distribution.

Using a standard normal distribution table or calculator:

- The p-value corresponding to a z-value of 0.2185 is approximately 0.4131.

### Step 3: Compare the P-value with the Significance Level
- **P-value**: 0.4131
- **Significance level (α)**: 0.10

Since the p-value (0.4131) is greater than the significance level (0.10), **we fail to reject the null hypothesis**.

### Conclusion:
There is not enough evidence at the 0.10 significance level to support the claim that the proportion of people who own cats is greater than 30%.

Would you like more details on any of these steps or further explanations?

Here are some related questions:
1. How do you interpret the p-value in this context?
2. What would happen if the significance level was lower, like 0.05?
3. How would the result change if the sample size was larger?
4. What are the assumptions behind using the z-test for proportions?
5. Can you explain the difference between a one-tailed and a two-tailed test?

**Tip:** Always ensure your sample size is large enough when performing hypothesis tests involving proportions to ensure the normal approximation is valid.

Learn how to test the claim that the proportion of people who own cats is larger than 30% using hypothesis testing. This problem involves calculating the z-test statistic, determining the p-value, and interpreting the results at a 0.10 significance level.

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