The problem presented involves calculating a regression equation to relate the monthly income (dependent variable) with years of service (independent variable). Here's how to approach each part of the question:

(a) Find the regression equation of income on years of service.

We are given:

• Years of Service (X): 11, 7, 9, 5, 8, 6, 10
• Income in Rs. 1000's (Y): 10, 8, 6, 5, 9, 7, 11

The linear regression equation is of the form: $Y = a + bX$ Where:

• $Y$ = predicted income
• $X$ = years of service
• $a$ = intercept
• $b$ = slope (rate of change of income with years of service)

Steps to find $a$ and $b$:

1. Calculate the means of $X$ (years of service) and $Y$ (income).
2. Compute the slope $b$ using the formula: $b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}$
3. Calculate the intercept $a$ using: $a = \bar{Y} - b\bar{X}$

Let's perform these calculations next.The regression equation for predicting income based on years of service is:

$Y = 2.0 + 0.75X$

Where:

• $Y$ is the predicted income in Rs. 1000s,
• $X$ is the years of service.

(b) Recommended Initial Start for 13 Years of Service

Using the regression equation $Y = 2.0 + 0.75X$, we can substitute $X = 13$ (years of service) to find the predicted income.

Let me calculate that next.The recommended initial income for a person applying with 13 years of service would be approximately Rs. 11,750 (since income is given in Rs. 1000's, this corresponds to 11.75 × 1000).

(c) Consideration of Other Factors

The regression model is based solely on years of service, but other factors could influence income. These might include:

1. Education and qualifications.
2. Job performance and experience in similar roles.
3. Industry standards and market conditions.
4. Skills and additional certifications.
5. Economic factors like inflation or company profitability.

Since regression focuses on one variable (years of service), adding more variables (multivariate regression) might yield a more accurate prediction.

Would you like more details or have any questions? Here are five related questions:

1. How do you calculate the coefficient of determination $R^2$ to measure the fit of the regression model?
2. How can other factors (such as skills) be included in a regression model?
3. What is the impact of outliers in a regression analysis?
4. How would you perform a hypothesis test for the slope $b$ to see if years of service significantly affects income?
5. How do we interpret the slope $b$ in practical terms?

Tip: For real-world applications, it's essential to verify that the relationship between variables is linear before relying on a linear regression model.