91Ó°ÊÓ

Explained variation refers to the portion of the total variation in the dependent variable (Weight, in this case) that can be explained by the independent variable (Overhead Width). It tells us how well our regression model performs in explaining the variations in the data.
To find the explained variation, we use the formula:
\[SS_{regression} = \sum{(\hat{y_i} - \bar{y})^2} \]
where:
\( \hat{y_i} \) = predicted value using the regression equation
\( \bar{y} \) = mean of the observed weights
In simpler terms, explained variation measures the differences between the predicted values of the regression model and the average of the observed values. The higher the explained variation, the better your model fits the data.

Unexplained variation, also known as residual variation, measures the discrepancy between the observed values and the predicted values from the regression model. It shows the amount of variation that the model fails to explain.
The formula to compute unexplained variation is:
\[SS_{residuals} = \sum{(y_i - \hat{y_i})^2} \]
where:
\( y_i \) = actual observed value
\( \hat{y_i} \) = predicted value from the regression equation
In other words, unexplained variation evaluates the scatter of data points around the regression line. A smaller unexplained variation indicates a more accurate model.

A prediction interval gives us a range within which we expect a new observation to fall, with a certain level of confidence. For example, a 99% prediction interval means that we are 99% confident that the actual value will lie within this range.
To calculate it, use the formula:
\[ \hat{y} \pm t_{\alpha/2,n-2} \sqrt{MS_{residual} (1 + \frac{1}{n} + \frac{(x_i - \bar{x})^2}{\sum{(x_i - \bar{x})^2}})} \]
where:
\( \hat{y} \) = predicted value
\( t_{\alpha/2,n-2} \) = t-score for the given confidence level and degrees of freedom
\( MS_{residual} \) = mean square error
The prediction interval considers both the variability in the data and the sample size to provide a more realistic range for future predictions. It's especially useful in practice, as it accounts for the natural uncertainty in predictions.

The least squares method is a standard approach for deriving the regression line in linear regression. It minimizes the sum of the squares of the residuals (the differences between observed and predicted values).
The regression equation is of the form:
\( Weight = a + b \times Overhead \ Width\)
To find the coefficients, you calculate the slope \( b \) and intercept \( a \) using the formulas:
\[ b = \frac{N \sum{xy} - \sum{x} \sum{y}}{N \sum{x^2} - (\sum{x})^2} \]
\[ a = \frac{\sum{y} - b \sum{x}}{N} \]
where:
\( N \) = number of data points
\( x \) = Overhead Width
\( y \) = Weight
By fitting the line through the data points such that the sum of the squared differences between observed and predicted values is minimized, we obtain the best-fit line for making predictions. This technique is fundamental in regression analysis and provides a basis for understanding the relationship between variables.