Question

a graph with x - axis labeled years since 1985 (ranging from 0 to 18) and y - axis labeled percent (with values 0, 7.33, 14.67, 22) shows a blue line. a regression equation for this line is $y = -0.42x + 19.26$. what do the slope and y - intercept indicate?
a. the slope indicates that the percent in 1985 should have been about 19%. the y - intercept indicates that the percent is falling at about 0.4% each year.
b. the slope indicates that the percent is falling at about 0.4% each year. the y - intercept indicates the percent in 1985 should have been about 19%.

Explanation:

To determine the meaning of the slope and y - intercept in the linear regression equation \(y=- 0.42x + 19.26\):

For the slope:

The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope. In the equation \(y=-0.42x + 19.26\), the slope \(m=- 0.42\). A negative slope indicates a decreasing relationship between the two variables (in this case, between the number of years since 1985 (\(x\)) and the percent (\(y\))). The magnitude of the slope (0.42) represents the rate of change of \(y\) with respect to \(x\). So, for each increase of 1 in \(x\) (each year since 1985), \(y\) (the percent) decreases by 0.42 (or about 0.4%) per year.

For the y - intercept:

The y - intercept \(b\) is the value of \(y\) when \(x = 0\). In the context of this problem, \(x\) represents the number of years since 1985. When \(x = 0\), it corresponds to the year 1985. Substituting \(x = 0\) into the equation \(y=-0.42x + 19.26\), we get \(y=19.26\approx19\). So the y - intercept represents the percent in the year 1985, which is about 19%.

Now let's analyze the options:

• Option a: It has the slope and y - intercept interpretations reversed. The slope is the rate of change (falling at about 0.4% per year) and the y - intercept is the value when \(x = 0\) (percent in 1985). So option a is incorrect.
• Option b: The slope interpretation (percent falling at about 0.4% each year) and the y - intercept interpretation (percent in 1985 should have been about 19%) are correct.

Answer:

b. The slope indicates that the percent is falling at about 0.4% each year. The y - intercept indicates the percent in 1985 should have been about 19%.