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Regression Analysis is a statistical technique that helps us understand the relationship between two variables. In our case, it involves analyzing the percentage of canine parvovirus (CPV) prevalence among adult wolves and pups (independent variable, \(x\)) and the resulting percentage of pups in capture (dependent variable, \(y\)).

The goal of regression analysis is to find the best-fit line, which we call the regression line, that minimizes the distance between the actual data points and the line itself. This line is described by the equation \( \hat{y} = b_0 + b_1x \), where \( \hat{y} \) is the predicted value, \( b_0 \) is the intercept and \( b_1 \) is the slope.

In this exercise, the equation is \( \hat{y} = 62.9476 - 0.54975x \), indicating that as CPV prevalence increases, the percentage of captured pups decreases, shown by the negative slope.

The Correlation Coefficient, denoted by \( r \), measures the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1.

A value close to 1 implies a strong positive relationship, while a value close to -1 indicates a strong negative relationship. A value around 0 suggests little or no linear relationship.

In this exercise, we have \( r^2 = 0.57 \) as the coefficient of determination. This means that 57% of the variability in the percentage of captured pups is explained by the percentage of CPV prevalence.

To find \( r \), we take the square root of \( r^2 \). The regression line slope is negative, indicating \( r = -\sqrt{0.57} \). This results in \( r = -0.755 \), confirming a moderate negative correlation, meaning that as CPV prevalence increases, the percentage of captured pups tends to decrease.

Residuals are the differences between the observed values and the values predicted by the regression line. The calculation of residuals is crucial because it helps to assess the goodness of the fit of the model.

You calculate a residual using the formula: \( \text{residual} = y - \hat{y} \), where \( y \) is the actual observed value and \( \hat{y} \) is the predicted value from the regression equation.

Using the observation \((25, 70)\), we find \( \hat{y} \) by substituting \( x = 25 \) into the regression equation:

• \( \hat{y} = 62.9476 - 0.54975 \times 25 = 49.71125 \)
• Residual = \( 70 - 49.71125 = 20.28875 \)

This positive residual indicates that the observed percentage of captured pups (70%) is higher than the predicted percentage (49.71125%), highlighting a deviation from the model's prediction at this point.

The Standard Error of the Estimate (\( s_e \)) is an important measure in regression analysis. It quantifies the typical distance that the observed values fall from the regression line.

A smaller standard error indicates that the data points are close to the regression line, signifying a better fit. Conversely, a larger standard error suggests more scatter around the line.

The formula for standard error is given by: \( s_e = \sqrt{\frac{SSTo - SSE}{n - 2}} \). Here, \( SSTo \) is the total sum of squares, and \( SSE \) is the sum of squared errors. For this exercise:

• Calculate \( SSE = r^2 \times SSTo = 0.57 \times 2520 = 1436.4 \)
• Then, calculate \( s_e = \sqrt{\frac{2520 - 1436.4}{10 - 2}} = 13.61 \)

The standard error of 13.61 suggests that, on average, the observed percentages of captured pups differ from the predicted values by about 13.61 percentage points.