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### Transcription of the Image:
**Question 4:**

Accounting for campaigns, one would investigate the effects of features (such as audience interest) on student performance (GCPA) of students at UTW.

1. **a.** In repeated campaigns at a fictitious company, specify the MLM fit for \( \mathbf{Y}_{ij} \). Also, model the above problem, drawing all the necessary variables. \[5 marks\]

2. **b.** Show how the average GCPA of students in the main, south and central campuses can be estimated from this specified model. \[5 marks\]

### Solution:

#### Part a: Specifying the Multilevel Model (MLM)

Let’s denote:
- \( \mathbf{Y}_{ij} \) as the performance (GCPA) of the \( i \)-th student in the \( j \)-th campaign.
- \( \beta_0 \) as the overall intercept.
- \( u_j \) as the random effect of the \( j \)-th campaign, representing the campaign-specific deviation.
- \( \epsilon_{ij} \) as the error term associated with the \( i \)-th student in the \( j \)-th campaign.
- \( X_{ij} \) as a vector of predictors (such as audience interest).

The MLM can be specified as:

\[
\mathbf{Y}_{ij} = \beta_0 + \mathbf{X}_{ij}\beta + u_j + \epsilon_{ij}
\]

Where:
- \( u_j \sim N(0, \sigma^2_u) \) (random effect with mean 0 and variance \( \sigma^2_u \)).
- \( \epsilon_{ij} \sim N(0, \sigma^2_\epsilon) \) (random error with mean 0 and variance \( \sigma^2_\epsilon \)).

This model accounts for the hierarchical structure of the data, where students are nested within campaigns.

#### Part b: Estimating Average GCPA for Different Campuses

Assume that the dataset includes a variable indicating the campus (e.g., "main," "south," "central"). Let’s denote the campus by \( C_j \) where \( C_j \in \{\text{main, south, central}\} \).

To estimate the average GCPA for students in each campus, we can modify the MLM to include fixed effects for the campuses:

\[
\mathbf{Y}_{ij} = \beta_0 + \beta_{\text{main}} I_{\text{main}}(C_j) + \beta_{\text{south}} I_{\text{south}}(C_j) + \beta_{\text{central}} I_{\text{central}}(C_j) + \mathbf{X}_{ij}\beta + u_j + \epsilon_{ij}
\]

Where:
- \( I_{\text{main}}(C_j) \) is an indicator variable that equals 1 if the \( j \)-th campaign is in the main campus and 0 otherwise.
- \( I_{\text{south}}(C_j) \) and \( I_{\text{central}}(C_j) \) are defined similarly for the south and central campuses.

The coefficients \( \beta_{\text{main}}, \beta_{\text{south}}, \beta_{\text{central}} \) would then give the expected difference in GCPA for students in these campuses relative to a reference campus (usually the one omitted from the model).

To find the average GCPA for students in each campus, take the model-predicted GCPA values for each \( C_j \) and average over all students in the campus.

### Summary:
1. **Model specification**: A multilevel model that accounts for hierarchical data structure, with students nested within campaigns.
2. **Fixed effects for campuses**: Adding fixed effects to estimate the impact of being in different campuses.
3. **Estimate average GCPA**: Use the fitted model to estimate the average GCPA for each campus by averaging predicted values.

### Do you have any questions or need further details? 
Here are 5 related questions that might interest you:

1. How do you determine if random effects are significant in a multilevel model?
2. What is the difference between fixed effects and random effects in multilevel modeling?
3. How can interaction terms be incorporated in this multilevel model?
4. What assumptions must be checked for the validity of this MLM?
5. How do you interpret the variance components in a multilevel model?

**Tip**: When working with hierarchical data, always ensure your model correctly reflects the structure of the data (e.g., nesting). This will lead to more accurate estimates and inferences.

Learn how to solve a multilevel modeling problem focusing on campaign effects and student performance at UTW. This detailed solution includes specifying a multilevel model, estimating average GCPA for different campuses, and interpreting results. Suitable for graduate-level students and researchers.

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