To conduct a hypothesis test using the critical value method, we’ll follow these steps:

Step 1: State the hypotheses

• Null Hypothesis (H₀): $p = 0.60$ (The true proportion of students who feel enriched is 60%.)
• Alternative Hypothesis (H₁): $p \neq 0.60$ (The true proportion of students who feel enriched is not 60%.)

This is a two-tailed test since we are testing whether the proportion is different from 60%, not specifically greater than or less than 60%.

Step 2: Determine the significance level

The significance level, $\alpha$, is given as 0.02 (or 2%).

Step 3: Calculate the test statistic

We’ll use the test statistic for proportions: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$ where:

• $\hat{p}$ is the sample proportion,
• $p_0$ is the hypothesized population proportion,
• $n$ is the sample size.

Given dataTo conduct a hypothesis test using the critical value method, we’ll follow these steps:

Step 1: State the hypotheses

• Null Hypothesis (H₀): $p = 0.60$ (The true proportion of students who feel enriched is 60%.)
• Alternative Hypothesis (H₁): $p \neq 0.60$ (The true proportion of students who feel enriched is not 60%.)

This is a two-tailed test since we are testing whether the proportion is different from 60%, not specifically greater than or less than 60%.

Step 2: Determine the significance level

The significance level, $\alpha$, is given as 0.02 (or 2%).

Step 3: Calculate the test statistic

We’ll use the test statistic for proportions: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$ where:

• $\hat{p}$ is the sample proportion,
• $p_0$ is the hypothesized population proportion,
• $n$ is the sample size.

Given

• $n = 64$ (sample size),
• $p_0 = 0.60$ (claimed proportion),
• $x = 30$ (number of students who felt enriched).

Calculate the sample proportion $\hat{p}$: $\hat{p} = \frac{x}{n} = \frac{30}{64} = 0.46875$

Now, calculate the standard deviation for the sampling distribution: $\sigma_{\hat{p}} = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.60 \times 0.40}{64}} = \sqrt{0.00375} \approx 0.06124$

Calculate the $z$-score: $z = \frac{\hat{p} - p_0}{\sigma_{\hat{p}}} = \frac{0.46875 - 0.60}{0.06124} \approx -2.14$

Step 4: Determine the critical value

Since we have a two-tailed test at a significance level of $\alpha = 0.02$, we’ll divide this by 2 to find the critical values for both tails.

• For $\alpha/2 = 0.01$, the critical $z$-values (from a standard normal table) are approximately $\pm 2.33$.

Step 5: Make a decision

• Decision rule: If $|z| > 2.33$, reject the null hypothesis.
• Test statistic: $z \approx -2.14$.

Since $|z| = 2.14$ is less than 2.33, we do not reject the null hypothesis.

Step 6: Conclusion

At the 2% level of significance, there is not enough evidence to reject the instructor's claim that 60% of her students feel more enriched.

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Would you like more details or have any questions?

Here are some related questions you might explore:

1. How does changing the sample size affect the result of the hypothesis test?
2. What happens if we use a different significance level, like 5% or 1%?
3. How would a one-tailed test differ in this scenario?
4. Why is it necessary to use a two-tailed test here?
5. How can the confidence interval help in interpreting the results of this test?

Tip: Always check if the sample size meets conditions for normal approximation in a hypothesis test for proportions.