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The problem involves comparing the book purchase behaviors of two groups: students from the Faculty of Commerce and students from the Faculty of Arts. There are two variables for each group:

- \( X_1 \): represents one type of book purchase behavior.
- \( X_2 \): represents another type of book purchase behavior.

The summary statistics provided include means (\( \bar{X} \)) and covariance matrices (\( S \)) for both faculties.

### Commerce Faculty:
- Mean: \( \bar{X}_C = [10.5, 31.2] \)
- Covariance matrix: 
  \[
  S_C = 
  \begin{bmatrix}
  4.3 & 9.8 \\
  9.8 & 34.2
  \end{bmatrix}
  \]

### Arts Faculty:
- Mean: \( \bar{X}_A = [8.5, 18.5] \)
- Covariance matrix: 
  \[
  S_A = 
  \begin{bmatrix}
  7.0 & 9.3 \\
  9.3 & 20.3
  \end{bmatrix}
  \]

The task is to determine if the book purchase behaviors (\( X_1 \) and \( X_2 \)) of Commerce and Arts students are significantly different using Bonferroni 95% simultaneous confidence intervals. To do this, we'll follow these steps:

### Steps:
1. **Formulate the Hypotheses**:
   - \( H_0 \): The means of the two faculties are equal, i.e., \( \bar{X}_C = \bar{X}_A \).
   - \( H_1 \): The means of the two faculties are significantly different.

2. **Bonferroni Confidence Intervals**:
   The Bonferroni method adjusts for multiple comparisons to control the family-wise error rate. Since we're testing two variables (\( X_1 \) and \( X_2 \)), we would calculate separate confidence intervals for each and apply the Bonferroni correction.

3. **Statistical Calculation**:
   We'll calculate the difference between the means for both \( X_1 \) and \( X_2 \), then compute the standard error and confidence intervals for each mean difference. Finally, we'll compare the confidence intervals to see if they include 0 (which would indicate no significant difference).

Let me know if you'd like me to go ahead and compute these intervals or if you'd like a further explanation of any of these steps.

### Relative Questions:
1. What are the assumptions behind the Bonferroni correction in hypothesis testing?
2. How are the covariance matrices used in calculating confidence intervals?
3. Why do we compare the means of two groups using confidence intervals?
4. What would the result imply if the confidence intervals do not contain zero?
5. How does the sample size affect the width of the confidence intervals?

**Tip:** When comparing multiple variables, corrections like Bonferroni help reduce the likelihood of falsely identifying significant differences.

Determine whether the book purchase behaviours (X1 and X2) of the two faculties (Commerce and Arts) are significantly different using Bonferroni 95% simultaneous confidence intervals. State the necessary assumptions.

Analyze the book purchase behaviors of Commerce and Arts students using Bonferroni 95% simultaneous confidence intervals. This problem involves hypothesis testing, covariance matrices, and the Bonferroni correction method. Suitable for undergraduate statistics students.

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