For question 3 in the image, we will calculate the coefficient of determination ($R^2$) and the standard deviation of the residuals for the relationship between age ($x$) and number of sick days ($y$).

1. Coefficient of Determination ($R^2$):

The coefficient of determination, $R^2$, tells us how well the regression line fits the data. It is calculated as:

$R^2 = 1 - \frac{SS_{\text{residual}}}{SS_{\text{total}}}$

Where:

• $SS_{\text{residual}}$ is the sum of the squared residuals: $\sum (y - \hat{y})^2$
• $SS_{\text{total}}$ is the total sum of squares: $\sum (y - \bar{y})^2$

2. Standard Deviation of the Residuals ($s_{\text{res}}$):

The standard deviation of the residuals is a measure of the typical size of the residuals (errors). It is given by:

$s_{\text{res}} = \sqrt{\frac{SS_{\text{residual}}}{n - 2}}$

Where:

• $n$ is the number of data points.

Let me now calculate these values based on the data you've provided in the image.Based on the data from question 3, the calculations are as follows:

1. Coefficient of Determination ($R^2$): $R^2 = -11.41$

   • This negative value indicates a very poor fit of the predicted values to the actual data. In practice, an $R^2$ value less than zero suggests that the model performs worse than simply using the mean of the data for prediction.

2. Standard Deviation of the Residuals: $s_{\text{res}} = 41.04$

   • This indicates a high variability in the residuals, meaning the predictions are quite far from the actual values on average.

Would you like a more detailed explanation of any of these results or clarifications on the calculations?

Here are 5 questions that expand on this topic:

1. How is the sum of squared residuals ($SS_{\text{residual}}$) used in assessing a model's fit?
2. What is the significance of having a negative $R^2$, and how should it be interpreted?
3. How does the standard deviation of residuals relate to the accuracy of a regression model?
4. How might the model be adjusted to improve the $R^2$ value?
5. What role do outliers play in increasing the residuals?

Tip: Always check the scatterplot of the data to visually assess the fit of a regression model, especially when $R^2$ suggests a poor fit.