Question

1. suppose that the scatterplot of (ln(x), ln(y))

following must be true?
i. the variables x and y also have a correlation close to 1.
ii. a scatterplot of (x, y) shows a strong nonlinear pattern.
iii. the residual plot of the variables x and y shows a random pattern.
(a) i only
(b) ii only
(c) iii only
(d) i and ii
(e) i, ii, and iii

1. which of the following is correct?

(a) the correlation r is the slope of the least-squares regression line.
(b) the square of the correlation is the slope of the least-squares regression line.
(c) the square of the correlation is the proportion of the data lying on the least-squares regression line.
(d) the coefficient of determination is the fraction of variability in y that can be explained by the least-squares
regression of y on x.
(e) the sum of the squared residuals from the least-squares line is 0.

1. which of the following statements concerning residuals from a lsrl is true?

(a) the sum of the residuals is always 0.
(b) a plot of the residuals is useful for assessing the fit of the least-squares regression line.
(c) the value of a residual is the observed value of the response minus the value of the response that once
would predict from the least-squares regression line.
(d) an influential point on a scatterplot is not necessarily the point with the largest residual.
(e) all of the above.

Explanation:

• For question 9:
• The correlation coefficient \(r\) measures the strength and direction of a linear relationship between two variables. It is not the slope of the least - squares regression line.
• The square of the correlation coefficient \(r^{2}\) (coefficient of determination) represents the proportion of the variability in \(y\) that can be explained by the least - squares regression of \(y\) on \(x\).
• The sum of the squared residuals from the least - squares line is not \(0\) in general (it is minimized, but not necessarily \(0\)).
• For question 10:
• By the properties of the least - squares regression line (LSRL), the sum of the residuals \(\sum_{i = 1}^{n}(y_{i}-\hat{y}_{i})=0\).
• A residual plot helps in assessing the fit of the LSRL. If the points in the residual plot are randomly scattered, the fit is good.
• The formula for a residual is \(e_{i}=y_{i}-\hat{y}_{i}\), where \(y_{i}\) is the observed value and \(\hat{y}_{i}\) is the predicted value from the LSRL.
• An influential point is a point that has a large effect on the regression line. It is not necessarily the point with the largest residual.

Answer:

• Question 9: D. The coefficient of determination is the fraction of variability in \(y\) that can be explained by the least - squares regression of \(y\) on \(x\).
• Question 10: E. All of the above.