Understanding Regression of Y on X for Commodity Prices

The question asks for the equation of the regression of Y on X, where X and Y represent prices of a commodity in Kolkata and Mumbai, respectively. This equation helps us predict the price in Mumbai (Y) based on the price in Kolkata (X).

We are given the following information:

• Mean price in Kolkata, \(\bar{X} = 65\) Rs
• Mean price in Mumbai, \(\bar{Y} = 67\) Rs
• Standard deviation of price in Kolkata, \(\sigma_X = 2.5\)
• Standard deviation of price in Mumbai, \(\sigma_Y = 3.5\)
• Correlation coefficient between X and Y, \(r(X, Y) = 0.8\)

Calculating the Regression Coefficient

The equation of the regression line of Y on X is given by \(Y = a + b_{YX}X\), where \(b_{YX}\) is the regression coefficient of Y on X and \(a\) is the y-intercept.

The formula for the regression coefficient \(b_{YX}\) is:

\(b_{YX} = r \frac{\sigma_Y}{\sigma_X}\)

Let's substitute the given values into the formula:

\(b_{YX} = 0.8 \times \frac{3.5}{2.5}\)

Calculating the value:

\(b_{YX} = 0.8 \times 1.4\)

\(b_{YX} = 1.12\)

So, the regression coefficient of Y on X is \(1.12\). This means that for every one-unit increase in the price in Kolkata (X), the price in Mumbai (Y) is expected to increase by \(1.12\) units, on average.

Calculating the Y-intercept

The regression line of Y on X always passes through the mean point \((\bar{X}, \bar{Y})\). We can use this property to find the y-intercept \(a\). The equation \(\bar{Y} = a + b_{YX}\bar{X}\) holds true.

We can rearrange this formula to solve for \(a\):

\(a = \bar{Y} - b_{YX}\bar{X}\)

Now, let's substitute the values of \(\bar{Y}\), \(b_{YX}\), and \(\bar{X}\):

\(a = 67 - (1.12 \times 65)\)

First, calculate the product \(1.12 \times 65\):

\(1.12 \times 65 = 72.8\)

Now, calculate \(a\):

\(a = 67 - 72.8\)

\(a = -5.8\)

The y-intercept is \(-5.8\). This is the predicted price in Mumbai when the price in Kolkata is 0 (though this interpretation may not be meaningful in the context of prices).

Forming the Regression Equation of Y on X

Now that we have the regression coefficient \(b_{YX} = 1.12\) and the y-intercept \(a = -5.8\), we can write the equation of the regression of Y on X:

\(Y = a + b_{YX}X\)

Substituting the calculated values:

\(Y = -5.8 + 1.12X\)

Rearranging the terms to match the standard form:

\(Y = 1.12X - 5.8\)

This is the equation that predicts the price in Mumbai (Y) based on the price in Kolkata (X).

Comparing with Given Options

Let's compare our derived equation, \(Y = 1.12X - 5.8\), with the given options:

• Option 1: \(Y = 0.175X - 5\)
• Option 2: \(Y = 1.12X - 5.8\)
• Option 3: \(Y = 1.12X - 5\)
• Option 4: \(Y = 0.17X + 5.8\)

Our calculated equation matches Option 2.

Revision Table: Key Regression Concepts

Additional Information on Linear Regression and Correlation

Linear regression is a statistical method used to model the relationship between a dependent variable (Y) and one or more independent variables (X). In simple linear regression, like this example, we have only one independent variable.

• Correlation vs. Regression: Correlation measures the degree and direction of the linear association between two variables, but it does not imply causation. Regression aims to model how the dependent variable changes as the independent variable changes, allowing for prediction.
• Properties of Regression Lines:
  • Both regression lines (Y on X and X on Y) pass through the point \((\bar{X}, \bar{Y})\).
  • The correlation coefficient \(r\) is the geometric mean of the two regression coefficients \(b_{YX}\) and \(b_{XY}\), i.e., \(r^2 = b_{YX} \times b_{XY}\). Also, the sign of \(r\) is the same as the sign of \(b_{YX}\) and \(b_{XY}\).
  • The closer the correlation coefficient \(|r|\) is to 1, the closer the data points cluster around the regression line, indicating a stronger linear relationship and better predictability.
• Applications: Regression analysis is widely used in economics, finance, social sciences, and many other fields to model relationships and make predictions. For example, predicting sales based on advertising expenditure, or predicting stock prices based on economic indicators.