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UPSC MainsMANAGEMENT-PAPER-II201316 Marks
Q3.

Question 3

In a class of 50 students, the regression equation of marks in Economics (X) on marks in Management (Y) is 3Y-5X+180 = 0. The mean marks in Management is 44 and the variance of marks in Economics is 9/16 of the variance of marks in Management. Find the mean marks in Economics and the coefficient of correlation between marks in the two subjects.

How to Approach

This question tests the understanding of linear regression and its associated calculations in statistics, a crucial component of quantitative analysis often used in management decision-making. The approach should involve first calculating the mean of X (Economics marks) using the regression equation. Then, utilizing the relationship between variances and the coefficient of correlation, the correlation coefficient should be determined. The solution requires a step-by-step application of statistical formulas.

Model Answer

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Introduction

Regression analysis is a powerful statistical tool used to examine the relationship between a dependent variable and one or more independent variables. In the context of educational assessment, it can be used to predict a student’s performance in one subject based on their performance in another. The regression equation provides a mathematical model describing this relationship. Understanding the regression coefficient and the correlation coefficient is vital for interpreting the strength and direction of this relationship. This question requires us to apply these concepts to determine the mean marks in Economics and the correlation between Economics and Management marks.

Calculating the Mean of Economics Marks (X)

The regression equation of X on Y is given as 3Y - 5X + 180 = 0. We can rewrite this equation to express X in terms of Y:

5X = 3Y + 180

X = (3/5)Y + 36

The mean of X (denoted as X̄) can be calculated using the formula:

X̄ = (3/5)Ȳ + 36

Given that the mean marks in Management (Ȳ) is 44, we can substitute this value into the equation:

X̄ = (3/5) * 44 + 36

X̄ = 26.4 + 36

X̄ = 62.4

Therefore, the mean marks in Economics is 62.4.

Calculating the Coefficient of Correlation (r)

Let σX be the standard deviation of Economics marks and σY be the standard deviation of Management marks. We are given that the variance of Economics marks (σX2) is 9/16 of the variance of Management marks (σY2):

σX2 = (9/16)σY2

Taking the square root of both sides, we get:

σX = (3/4)σY

The regression coefficient (bXY) of X on Y is given by bXY = 3/5 = 0.6. The formula for the regression coefficient is:

bXY = r * (σX/σY)

Substituting the values we have:

0.6 = r * ( (3/4)σY / σY)

0.6 = r * (3/4)

r = 0.6 * (4/3)

r = 0.8

Therefore, the coefficient of correlation between marks in the two subjects is 0.8.

Summary of Results

Parameter Value
Mean Marks in Economics (X̄) 62.4
Coefficient of Correlation (r) 0.8

Conclusion

In conclusion, we have successfully determined the mean marks in Economics to be 62.4 and the coefficient of correlation between Economics and Management marks to be 0.8. This indicates a strong positive correlation, suggesting that students who perform well in Management tend to perform well in Economics as well. The regression analysis provides valuable insights into the relationship between these two subjects, which can be used for academic advising and curriculum development.

Answer Length

This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.

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Additional Resources

Key Definitions

Regression Analysis
A statistical method used to determine the relationship between a dependent variable and one or more independent variables. It helps in predicting the value of the dependent variable based on the values of the independent variables.
Coefficient of Correlation
A statistical measure that expresses the extent to which two variables are linearly related. It ranges from -1 to +1, where +1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no linear correlation.

Key Statistics

The average correlation between SAT scores and first-year college GPA is around 0.4 to 0.5 (source: National Association for College Admission Counseling, 2023).

Source: National Association for College Admission Counseling (2023)

In 2022, India's Gross Enrolment Ratio (GER) in higher education was 27.3%, indicating the proportion of eligible population enrolled in higher education (source: All India Survey on Higher Education, 2022-23).

Source: All India Survey on Higher Education (2022-23)

Examples

Predicting Sales

A company might use regression analysis to predict future sales based on advertising expenditure. The advertising expenditure would be the independent variable, and sales would be the dependent variable.

Frequently Asked Questions

▶What does a correlation coefficient of 0.8 indicate?

A correlation coefficient of 0.8 indicates a strong positive correlation. This means that as one variable increases, the other variable tends to increase as well, and the relationship is relatively strong.

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