(a)
11.0

Chi-squared test for independence

Expected frequency: \( E = \frac{\text{row total} \times \text{column total}}{\text{grand total}} \)

Breakfast: Perfect \( \frac{232 \times 204}{500} = 94.656 \), Satisfactory \( \frac{232 \times 274}{500} = 127.184 \), Poor \( \frac{232 \times 22}{500} = 10.208 \)

Lunch: Perfect \( \frac{154 \times 204}{500} = 62.832 \), Satisfactory \( \frac{154 \times 274}{500} = 84.392 \), Poor \( \frac{154 \times 22}{500} = 6.776 \)

Dinner: Perfect \( \frac{114 \times 204}{500} = 46.512 \), Satisfactory \( \frac{114 \times 274}{500} = 62.424 \), Poor \( \frac{114 \times 22}{500} = 5.016 \)

\( \chi^2 = \sum \frac{(O – E)^2}{E} \)

Contributions: Breakfast, Perfect \( \frac{(101 – 94.656)^2}{94.656} \approx 0.4246 \)

Satisfactory \( \frac{(124 – 127.184)^2}{127.184} \approx 0.0797 \), Poor \( \frac{(7 – 10.208)^2}{10.208} \approx 1.0072 \)

Lunch, Perfect \( \frac{(68 – 62.832)^2}{62.832} \approx 0.4248 \), Satisfactory \( \frac{(81 – 84.392)^2}{84.392} \approx 0.1364 \)

Poor \( \frac{(5 – 6.776)^2}{6.776} \approx 0.4656 \)

Dinner, Perfect \( \frac{(35 – 46.512)^2}{46.512} \approx 2.8519 \), Satisfactory \( \frac{(69 – 62.424)^2}{62.424} \approx 0.6926 \)

Poor \( \frac{(10 – 5.016)^2}{5.016} \approx 4.9504 \)

Sum: \( \approx 0.4246 + 0.0797 + 1.0072 + 0.4248 + 0.1364 + 0.4656 + 2.8519 + 0.6926 + 4.9504 \approx 11.0332 \)

Rounded to three significant figures: 11.0

Result: 11.0 [2]

(b)
Reject \( H_0 \), quality of food and type of meal are not independent

Chi-squared test for independence, \( df = (3-1)(3-1) = 4 \)

Significance level: \( \alpha = 0.05 \)

Critical value: 9.488

Method 1
\( \chi^2 \approx 11.0 > 9.488 \)

Reject \( H_0 \)

Method 2
p-value \( \approx 0.0263 \)

\( p < 0.05 \), reject \( H_0 \)

Conclusion: Evidence suggests quality and meal type are dependent

Result: Reject \( H_0 \), not independent [2]