To determine if the class distribution for theatergoers is significantly different from the general college population, we conduct a chi-square goodness-of-fit test.

Step 1: State the hypotheses

• Null hypothesis (H₀): The distribution of theatergoers' class status matches the distribution of the general college population.
• Alternative hypothesis (H₁): The distribution of theatergoers' class status is significantly different from the general college population.

Step 2: Calculate expected frequencies

The total number of theatergoers = $20 + 23 + 22 + 15 = 80$.
Use the college population proportions to compute the expected counts:

• Freshmen: $0.30 \times 80 = 24$
• Sophomores: $0.25 \times 80 = 20$
• Juniors: $0.25 \times 80 = 20$
• Seniors: $0.20 \times 80 = 16$

Step 3: Compute the chi-square statistic

The chi-square statistic is calculated using:

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

Where:

• $O_i$ = observed frequencies
• $E_i$ = expected frequencies

Class Status	Observed ($O_i$)	Expected ($E_i$)	$(O_i - E_i)^2 / E_i$
Freshmen	20	24	$\frac{(20 - 24)^2}{24} = \frac{16}{24} = 0.667$
Sophomores	23	20	$\frac{(23 - 20)^2}{20} = \frac{9}{20} = 0.450$
Juniors	22	20	$\frac{(22 - 20)^2}{20} = \frac{4}{20} = 0.200$
Seniors	15	16	$\frac{(15 - 16)^2}{16} = \frac{1}{16} = 0.0625$

$\chi^2 = 0.667 + 0.450 + 0.200 + 0.0625 = 1.3795$

Rounded to two decimal places: $\chi^2 = 1.38$.

Step 4: Determine the critical value

The degrees of freedom (df) = $\text{number of categories} - 1 = 4 - 1 = 3$.
At the $\alpha = 0.05$ significance level, the critical value from the chi-square table is:

$\chi^2_{\text{critical}} = 7.815$

Step 5: Compare and conclude

• $\chi^2 = 1.38$
• $\chi^2_{\text{critical}} = 7.815$

Since $\chi^2 = 1.38 < 7.815$, we fail to reject the null hypothesis.

Conclusion

The distribution for theatergoers is not significantly different from the distribution for the general college population.

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Summary

• $\chi^2_{\text{critical}} = 7.815$
• $\chi^2 = 1.38$
• Fail to reject the null hypothesis. The distribution for theatergoers is not significantly different from the general student population.

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Do you want further clarification or detailed breakdowns of any step?
Here are 5 related questions:

1. How is the chi-square critical value determined?
2. What are the assumptions of the chi-square goodness-of-fit test?
3. How does changing the significance level affect the conclusion?
4. How would the result change if the observed frequencies were more extreme?
5. How is the degrees of freedom calculated?

Tip: Always compare $\chi^2$ with the critical value to decide on the hypothesis.