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UPSC MainsMANAGEMENT-PAPER-I202010 Marks
Q9.

Based on the information, determine the least squared linear regression model.

How to Approach

This question requires applying statistical methods, specifically linear regression. The approach involves understanding the concept of least squares, the linear regression equation, and how to determine the coefficients (slope and intercept) that minimize the sum of squared errors. The answer should demonstrate the ability to apply this mathematical concept, even without provided data. Since no data is given, the answer will focus on the *method* of determining the least squared linear regression model, outlining the steps and formulas involved.

Model Answer

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Introduction

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. The goal is to find the best-fitting line that describes this relationship. The 'least squares' method is a standard approach to estimate the parameters of this line – the slope and the intercept – by minimizing the sum of the squared differences between the observed and predicted values. This technique is widely used in various fields, including economics, finance, and engineering, to predict future outcomes and understand underlying trends. Determining this model involves a series of calculations based on the available data points.

Understanding the Linear Regression Model

The basic linear regression model can be represented as:

Y = β0 + β1X + ε

Where:

  • Y is the dependent variable
  • X is the independent variable
  • β0 is the intercept (the value of Y when X = 0)
  • β1 is the slope (the change in Y for a one-unit change in X)
  • ε is the error term (representing the difference between the observed and predicted values)

The Least Squares Method

The least squares method aims to find the values of β0 and β1 that minimize the sum of squared errors (SSE). The SSE is calculated as:

SSE = Σ(Yi - Ŷi)2

Where:

  • Yi is the observed value of the dependent variable for the i-th observation
  • Ŷi is the predicted value of the dependent variable for the i-th observation (Ŷi = β0 + β1Xi)
  • Σ denotes the summation over all observations

Calculating the Slope (β1)

The formula for calculating the slope (β1) using the least squares method is:

β1 = Σ[(Xi - X̄)(Yi - Ȳ)] / Σ[(Xi - X̄)2]

Where:

  • X̄ is the mean of the independent variable
  • Ȳ is the mean of the dependent variable

Calculating the Intercept (β0)

Once the slope (β1) is calculated, the intercept (β0) can be calculated using the following formula:

β0 = Ȳ - β1X̄

Steps to Determine the Least Squared Linear Regression Model

  1. Collect Data: Gather the data points for the independent (X) and dependent (Y) variables.
  2. Calculate Means: Calculate the mean of the independent variable (X̄) and the mean of the dependent variable (Ȳ).
  3. Calculate Deviations: For each data point, calculate the deviation of X from its mean (Xi - X̄) and the deviation of Y from its mean (Yi - Ȳ).
  4. Calculate the Slope: Use the formula for β1 to calculate the slope of the regression line.
  5. Calculate the Intercept: Use the formula for β0 to calculate the intercept of the regression line.
  6. Formulate the Equation: Substitute the calculated values of β0 and β1 into the linear regression equation (Y = β0 + β1X) to obtain the least squared linear regression model.

Example (Illustrative - No Data Provided)

Let's assume, hypothetically, after performing the calculations with a dataset, we find:

  • β0 = 2.5
  • β1 = 1.2

Then, the least squared linear regression model would be:

Y = 2.5 + 1.2X

Conclusion

Determining the least squared linear regression model involves a systematic application of statistical formulas to minimize the sum of squared errors. While this explanation outlines the process, it's crucial to have a dataset to perform the actual calculations. The resulting equation provides a predictive model that can be used to estimate the value of the dependent variable based on the value of the independent variable. This technique is a fundamental tool in data analysis and decision-making across numerous disciplines.

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 process for estimating the relationships among variables. It includes many techniques for modeling the relationship between a dependent variable and one or more independent variables.
Sum of Squared Errors (SSE)
A measure of the difference between the actual values and the values predicted by a model. It is calculated by summing the squares of the residuals (the differences between observed and predicted values).

Key Statistics

According to Statista, the global market size of regression analysis software was valued at approximately 1.8 billion USD in 2023.

Source: Statista (as of knowledge cutoff 2023)

A study by McKinsey found that companies that are data-driven are 23 times more likely to acquire customers and 6 times more likely to retain them.

Source: McKinsey Global Institute (as of knowledge cutoff 2023)

Examples

House Price Prediction

Predicting the price of a house based on its size (square footage) using linear regression. The size of the house would be the independent variable, and the price would be the dependent variable.

Frequently Asked Questions

▶What does R-squared tell us about the model?

R-squared (Coefficient of Determination) represents the proportion of variance in the dependent variable that is predictable from the independent variable(s). A higher R-squared value indicates a better fit of the model to the data.

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