Using the table of units sold and corresponding sales for 8 branches of a company, calculate the regression equation, predict sales for a unit sale, compute the Sum of Squares of E... Using the table of units sold and corresponding sales for 8 branches of a company, calculate the regression equation, predict sales for a unit sale, compute the Sum of Squares of Errors (SSE), determine the Coefficient of Determination (R²), and calculate the Standard Error of Estimate (SEE). Read more

Steps to Solve

1. Calculate the Regression Equation

To calculate the regression equation, we first need to determine the slope ($m$) and y-intercept ($b$) using the formulas:

The slope is calculated by:

$$ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} $$

The y-intercept is calculated using:

$$ b = \frac{\sum y - m(\sum x)}{N} $$

Where $N$ is the number of data points, $x$ is the units sold, $y$ is the sales, $\sum xy$ is the sum of the product of $x$ and $y$, $\sum x$ is the sum of all $x$ values, $\sum y$ is the sum of all $y$ values, $\sum x^2$ is the sum of the square of all $x$ values.

2. Make Predictions

Once we have the regression equation in the form $y = mx + b$, we can use this equation to make predictions. Substitute the value of $x$ (units sold) into the equation to find the predicted sales ($\hat{y}$):

$$ \hat{y} = mx + b $$

3. Calculate the Sum of Squares of Errors (SSE)

To evaluate the model, you will compute the Sum of Squares of Errors (SSE):

$$ SSE = \sum (y_i - \hat{y}_i)^2 $$

Where $y_i$ is the actual sales, and $\hat{y}_i$ is the predicted sales from the regression equation.

4. Calculate Coefficient of Determination (R²)

Finally, the Coefficient of Determination (R²) can be found using:

$$ R^2 = 1 - \frac{SSE}{SST} $$

Where $SST$ is the total sum of squares, calculated as:

$$ SST = \sum (y_i - \bar{y})^2 $$

Here, $\bar{y}$ is the mean of the actual sales.