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Given the data Determine the 95% confidence interval for the population partial coefficients for Rank and Hourly. Do you think the firm’s rank and the annual salary for hourly employees will significantly affect the annual salary for salaried employees? 

To determine the 95% confidence interval for the population partial coefficients for Rank and Hourly, we'll need to fit a multiple linear regression model using the data provided. The goal is to see how Rank and Hourly (independent variables) predict Salaried (dependent variable). From there, we can estimate the coefficients and calculate the confidence intervals.

Here’s the general procedure I will follow:

1. **Fit the regression model**: We will run a multiple linear regression using Salaried as the dependent variable and Rank (X1) and Hourly (X2) as the independent variables.
2. **Obtain the regression coefficients**: From the output, I will extract the coefficients for Rank and Hourly.
3. **Compute the 95% confidence intervals**: Using the standard error of the coefficients and the t-distribution, we will calculate the confidence intervals for Rank and Hourly.
4. **Assess significance**: We will check the p-values for the coefficients to determine if Rank and Hourly significantly affect Salaried.

Let me perform the regression and calculations. I will proceed with coding that analysis now.It seems like there was a mismatch in the lengths of the data columns, which caused an error. I'll fix the issue and rerun the analysis by making sure that all the columns have the same number of entries.

Let me correct the dataset and proceed with the regression analysis again.Here are the results of the multiple linear regression:

- **Coefficients**:
  - Intercept (constant): 83.28
  - Rank (X1): -0.09
  - Hourly (X2): 0.44

- **P-values**:
  - Intercept: 0.0008 (statistically significant)
  - Rank (X1): 0.6771 (not statistically significant)
  - Hourly (X2): 0.2524 (not statistically significant)

- **95% Confidence Intervals**:
  - Intercept: [37.88, 128.69]
  - Rank (X1): [-0.55, 0.36]
  - Hourly (X2): [-0.33, 1.20]

### Interpretation:
- The **Rank (X1)** coefficient has a p-value of 0.6771, indicating that it is **not statistically significant** at the 95% confidence level. The confidence interval also crosses zero (from -0.55 to 0.36), suggesting no strong evidence that Rank affects the salaried income.
  
- The **Hourly (X2)** coefficient has a p-value of 0.2524, which also indicates it is **not statistically significant**. The confidence interval ranges from -0.33 to 1.20, meaning that there is no clear evidence that the hourly rate has a significant impact on salaried income.

### Conclusion:
Based on this analysis, neither the firm’s rank nor the annual salary for hourly employees **significantly** affects the annual salary for salaried employees at the 95% confidence level.

Would you like more details or have any further questions?

Here are some related questions for exploration:
1. What happens if you include more predictors like company size?
2. How would the confidence intervals change with more data points?
3. Is there a nonlinear relationship between Rank and Salaried?
4. Could interaction terms between Rank and Hourly provide more insight?
5. What are the implications of excluding outliers from the analysis?

**Tip**: Always check the p-values and confidence intervals together to assess the significance and reliability of regression coefficients.

Given the data, determine the 95% confidence interval for the population partial coefficients for Rank and Hourly. Do you think the firm’s rank and the annual salary for hourly employees will significantly affect the annual salary for salaried employees?

This problem involves calculating 95% confidence intervals for regression coefficients and assessing the significance of Rank and Hourly variables. Using a multiple linear regression model, the solution evaluates if Rank and the hourly wage significantly affect the salaried income. Suitable for students at the undergraduate or graduate level.

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