To find the test statistic and critical value, we need to first establish the null hypothesis (H0) and alternative hypothesis (H1) regarding the proportion of cat owners in the population.
Let's assume:
- Null Hypothesis (H0): p = 0.50 (the population proportion of cat owners is 50%).
- Alternative Hypothesis (H1): p > 0.50 (the population proportion of cat owners is greater than 50%).
Step 1: Calculate the test statistic
We will use the formula for the z-test statistic for proportions:
\[ z = \frac{(\hat{p} - p_0)}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \]
Where:
- \(\hat{p} = 0.83\) (sample proportion)
- \(p_0 = 0.50\) (hypothesized population proportion)
- \(n = 100\) (sample size)
Now plug in the values:
\[ z = \frac{(0.83 - 0.50)}{\sqrt{\frac{0.50 \cdot (1 - 0.50)}{100}}} \] \[ z = \frac{0.33}{\sqrt{\frac{0.50 \cdot 0.50}{100}}} \] \[ z = \frac{0.33}{\sqrt{\frac{0.25}{100}}} \] \[ z = \frac{0.33}{\sqrt{0.0025}} \] \[ z = \frac{0.33}{0.05} \] \[ z = 6.6 \]
Step 2: Determine the critical value
For a one-tailed test at a 0.05 significance level, you can find the critical z-value using standard normal distribution tables or a z-score calculator. The critical z-value (z-critical) for a significance level of 0.05 in a right-tailed test is approximately:
\[ z_{\text{critical}} \approx 1.645 \]
Summary
Based on the calculations:
- The test statistic is 6.60 (to 2 decimals).
- The critical value is 1.65 (to 2 decimals).
Thus, we can conclude that since the test statistic (6.60) is greater than the critical value (1.65), we would reject the null hypothesis in favor of the alternative hypothesis.
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