Understanding Sales and Advertisement Expenditure Regression

This problem asks us to predict the likely sales of a firm based on a proposed advertisement expenditure. We are given data including the means and standard deviations of sales and advertisement expenditure, as well as the correlation coefficient between them. This information suggests that we can use linear regression analysis to find the relationship between these two variables and make a prediction.

Given Data for Sales and Advertisement

Let Sales be denoted by \(S\) and Advertisement Expenditure be denoted by \(A\). The given data is:

We are also given the correlation coefficient between Sales and Advertisement Expenditure:

• Correlation coefficient, \(r = 0.9\)

We need to find the likely sales (\(S\)) when the advertisement expenditure (\(A\)) is Rs. 10 crore.

Applying Linear Regression for Sales Prediction

Since we want to predict Sales (\(S\)) based on Advertisement Expenditure (\(A\)), we need to find the regression line of \(S\) on \(A\). The general equation for the regression line of Y on X is:

\(Y - \bar{Y} = r \frac{\sigma_Y}{\sigma_X} (X - \bar{X})\)

In our case, Y is Sales (\(S\)) and X is Advertisement Expenditure (\(A\)). So, the equation for the regression line of \(S\) on \(A\) is:

\(S - \bar{S} = r \frac{\sigma_S}{\sigma_A} (A - \bar{A})\)

Calculating the Regression Coefficients

First, let's calculate the slope of the regression line of \(S\) on \(A\), denoted as \(b_{SA}\):

\(b_{SA} = r \frac{\sigma_S}{\sigma_A}\)

Substitute the given values:

\(b_{SA} = 0.9 \times \frac{10}{1.5}\)

To simplify the calculation:

\(b_{SA} = 0.9 \times \frac{100}{15} = \frac{9}{10} \times \frac{100}{15} = \frac{9 \times 10}{15} = \frac{90}{15}\)

\(b_{SA} = 6\)

Now, we can write the complete regression equation of \(S\) on \(A\):

\(S - \bar{S} = b_{SA} (A - \bar{A})\)

Substitute the means (\(\bar{S}=40\), \(\bar{A}=6\)) and the calculated slope (\(b_{SA}=6\)):

\(S - 40 = 6 (A - 6)\)

Predicting Sales for Rs. 10 Crore Advertisement

We need to find the likely sales (\(S\)) when the advertisement expenditure (\(A\)) is Rs. 10 crore. Substitute \(A = 10\) into the regression equation:

\(S - 40 = 6 (10 - 6)\)

\(S - 40 = 6 (4)\)

\(S - 40 = 24\)

Now, solve for \(S\):

\(S = 24 + 40\)

\(S = 64\)

So, the likely sales for a proposed advertisement expenditure of Rs. 10 crore is Rs. 64 crores.

Final Answer Derivation

Based on the linear regression analysis using the given data, the predicted sales figure is Rs. 64 crores.

Revision Table: Sales and Advertisement Regression

Additional Information on Regression Analysis

Regression analysis is a powerful statistical tool used in various fields, including business and economics, to understand how variables relate to each other. In this problem, we used simple linear regression because we were analyzing the relationship between two variables: sales and advertisement expenditure. There are other types of regression, such as multiple linear regression (involving more than one independent variable) and non-linear regression.

The correlation coefficient (\(r\)) tells us about the strength and direction of the linear association. A value of 0.9 indicates a strong positive linear relationship between advertisement expenditure and sales, meaning that as advertisement expenditure increases, sales tend to increase significantly.

The regression line provides the best linear estimate of the dependent variable (Sales) based on the independent variable (Advertisement Expenditure). The calculated slope of 6 means that, on average, for every one crore of Rs. increase in advertisement expenditure, sales are expected to increase by 6 crore of Rs.

Using the regression equation, we can make predictions within the range of the original data. Extrapolating predictions far outside the original data range (e.g., predicting sales for an advertisement expenditure of Rs. 100 crore based on data with maximum expenditure around Rs. 6 crore) might not be reliable as the relationship might not remain linear.