91Ó°ÊÓ

Residuals are important in linear regression because they show how far the actual data points deviate from the predicted values on the regression line. These deviations are expressed as differences. To calculate a residual, subtract the predicted value from the actual observed value. In this exercise, we have the equation of a regression line:

• \( \hat{y} = 62.9476 - 0.54975x \)

Let's take an observation with actual values of \(x = 25\) and \(y = 70\). We substitute \(x = 25\) into the equation to find the predicted \(\hat{y}\). Hence, the predicted value \(\hat{y}\) is:

• \( \hat{y} = 62.9476 - 0.54975 \times 25 = 49.71 \)

The residual is calculated by subtracting this predicted value from the actual observed value, \(y = 70\). Therefore, the residual is:

• \( y - \hat{y} = 70 - 49.71 = 20.29 \)

Understanding residuals helps to improve the regression model by identifying potential outliers or suggesting modifications to the regression model, such as changing the model form or removing outliers.

The correlation coefficient, denoted by \(r\), measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where values close to 1 or -1 indicate a strong relationship, and a value close to 0 suggests a weak linear relationship. In this exercise, we start with the coefficient of determination, \(r^2 = 0.57\), which tells us that 57% of the variability in the percentage of pups captured can be explained by the prevalence of CPV.To find \(r\), we take the square root of \(r^2\). Given \(r^2 = 0.57\), then:

• \( r = \pm \sqrt{0.57} \)
• \( r \approx \pm 0.75 \)

Since the slope of the regression line is negative, indicating that as the percentage of CPV increases, the percentage of captured pups decreases, we use the negative root. Hence, \(r = -0.75\), revealing a moderate negative correlation. This means as the CPV prevalence increases, there is a tendency for the pup capture percentage to decrease.

The standard error of regression is a measure of the accuracy with which a regression line predicts the data points. It provides an estimate of the standard deviation of the residuals or how much the observed values differ from the predicted values, on average. To compute the standard error \(s_{e}\), we use the residual sum of squares (SSR) and divide it by the degrees of freedom, take its square root.We know:

• The total sum of squares \(SSTo = 2520.0\)
• The coefficient of determination \(r^2 = 0.57\)

First, calculate the residual sum of squares (SSR) using the formula:

• \( SSR = SSTo \times (1- r^2) = 2520.0 \times (1 - 0.57) = 1084.40 \)

With 10 observations, the degrees of freedom are \(n - 2 = 8\). Finally, the standard error \(s_{e}\) is computed as:

• \( s_{e} = \sqrt{\frac{SSR}{\text{degrees of freedom}}} = \sqrt{\frac{1084.40}{8}} \approx 11.67 \)

The standard error \(s_{e}\) quantifies the typical distance that observed responses deviate from the regression line, providing insight into the model’s predictive precision.