To evaluate the instructor's claim, we can perform a hypothesis test for a population proportion. Here are the steps to determine whether the sample data supports the instructor's claim that 60% of her students feel enriched.

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Step 1: Formulate the Hypotheses

• Null Hypothesis ($H_0$): $p = 0.60$
  (The proportion of students who feel enriched is 60%.)
• Alternative Hypothesis ($H_a$): $p \neq 0.60$
  (The proportion of students who feel enriched is different from 60%.)

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Step 2: Check the Sample Data

• Sample size ($n$) = 64
• Number of students feeling enriched ($x$) = 34
• Sample proportion ($\hat{p}$) = $\frac{x}{n} = \frac{34}{64} = 0.53125$.

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Step 3: Determine the Test Statistic

We will use the z-test for proportions:

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

Where:

• $p_0 = 0.60$ (hypothesized population proportion)
• $\hat{p} = 0.53125$ (sample proportion)

Substituting values:

$z = \frac{0.53125 - 0.60}{\sqrt{\frac{0.60(1-0.60)}{64}}}$

$z = \frac{-0.06875}{\sqrt{\frac{0.24}{64}}}$

$z = \frac{-0.06875}{0.06125} \approx -1.12$

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Step 4: Determine the Critical Value and p-value

For a two-tailed test at a 5% significance level ($\alpha = 0.05$), the critical z-values are approximately $\pm 1.96$.

• Using standard z-tables, the p-value for $z = -1.12$ is approximately 0.263.

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Step 5: Make a Decision

• Since $z = -1.12$ falls within the range $[-1.96, 1.96]$, and the p-value (0.263) is greater than $\alpha = 0.05$, we fail to reject the null hypothesis.

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Conclusion

The data does not provide sufficient evidence to conclude that the true proportion of students feeling enriched is different from 60%. The instructor's claim could be reasonable based on this analysis.

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Do you want further details or have any questions?
Here are some related questions for exploration:

1. What is the meaning of a p-value in hypothesis testing?
2. How would the conclusion change if the sample size were larger?
3. What would happen if we conducted a one-tailed test instead?
4. How does changing the significance level ($\alpha$) affect the results?
5. Could other factors, like survey bias, influence these results?

Tip: Always check if the sample size is large enough to justify using normal approximation for a proportion test. For this test, $n \cdot p_0 \geq 10$ and $n \cdot (1 - p_0) \geq 10$ should hold true!