Question

Based on a sample of 100 people, 83% owned cats

The test statistic is:
6.60
Incorrect (to 2 decimals)

The critical value is:
1.65
Incorrect (to 2 decimals)

Answer 1

To conduct a hypothesis test regarding the proportion of cat owners in a sample of 100 people, we can outline the steps to find the test statistic and critical value. Let's assume we want to test the null hypothesis \(H_0: p = 0.5\) (i.e., 50% ownership) against the alternative hypothesis \(H_a: p > 0.5\).

### Step 1: Identify sample proportion (p̂)

Given:
- Sample size (n) = 100
- Proportion owning cats (p̂) = 0.83

### Step 2: Calculate the Test Statistic

We use the formula for the test statistic for proportions:

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

Where:
- \( \hat{p} \) = sample proportion (0.83)
- \( p_0 \) = hypothesized population proportion (0.5)
- \( n \) = sample size (100)

Now we can calculate it:

1. **Calculate the standard error**:

\[
SE = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.5 (1 - 0.5)}{100}} = \sqrt{\frac{0.5 \cdot 0.5}{100}} = \sqrt{\frac{0.25}{100}} = \sqrt{0.0025} = 0.05
\]

2. **Calculate the test statistic (z)**:

\[
z = \frac{0.83 - 0.5}{0.05} = \frac{0.33}{0.05} = 6.6
\]

### Conclusion on Test Statistic
The test statistic is 6.60, which matches your original statistic, but you mentioned it was incorrect. Please recheck the assumptions and hypotheses of your test.

### Step 3: Critical Value

Using a z-table or standard normal distribution for a one-tailed test at the usual significance level of 0.05, the critical value (z*) is approximately:

\[
z_{0.05} \approx 1.645 \text{ (You previously stated 1.65)}
\]

### Summary:
- **Test Statistic (z)**: 6.60
- **Critical Value**: For a one-tailed test at \(\alpha = 0.05\), the value is approximately 1.645. If you're using a rounded approach, perhaps 1.65 is acceptable, but often it's better to be precise with the critical value.

If you have a different significance level or a two-tailed test, the critical value could differ. Please provide additional context if adjustments are necessary.