Question

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

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:

• Question 9:
• The correlation \(r\) is not the slope of the least - squares regression line. The slope \(b = r\frac{s_y}{s_x}\).
• The square of the correlation (\(r^{2}\)) is not the slope.
• The square of the correlation (\(r^{2}\)) is the coefficient of determination, which is the fraction of 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 \(SSE=\sum(y_i-\hat{y}_i)^{2}\), and it is not \(0\) unless all data points lie on the line.
• Question 10:
• For a least - squares regression line (LSRL), \(\sum_{i = 1}^{n}(y_i-\hat{y}_i)=0\) (by the property of the least - squares method).
• A residual plot (\(y_i-\hat{y}_i\) vs \(x_i\)) helps in assessing the fit. If the plot shows a random pattern, the linear model is a good fit.
• The residual \(e_i=y_i - \hat{y}_i\) (observed minus predicted).
• 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
• Question 10: E