Question

how could finding the y - intercept of a trend line for a data set help determine the usefulness of the trend line as a model?
choose the correct answer below.
a. the y - intercept of all trend lines is 0. if the y - intercept is not 0, then a linear model does not make sense for the data.
b. it is not possible to construct the equation of the trend line without knowing the y - intercept.
c. the y - intercept gives the initial value at x = 0, and interpreting this value can help determine whether a linear model makes sense.
d. the y - intercept can be used to determine whether the association of the data is positive or negative.

Explanation:

• Option A: The y - intercept of a trend line is not always 0. For example, in the equation \(y = 2x+3\), the y - intercept is 3. So, this option is incorrect.
• Option B: The equation of a trend line (in the form \(y=mx + b\)) can be constructed using methods like the least - squares method where we first find the slope \(m\) and then solve for \(b\) (the y - intercept). But we can also write the point - slope form \(y - y_1=m(x - x_1)\) without explicitly knowing \(b\) at first. So, this option is incorrect.
• Option C: In a linear model \(y=mx + b\), when \(x = 0\), \(y=b\) (the y - intercept). If, for example, \(x\) represents time (in years) and \(y\) represents the number of customers, and \(b\) is a negative number when the number of customers cannot be negative at \(x = 0\) (the start time), then the linear model may not be a good fit. Interpreting the y - intercept (\(b\)) helps in evaluating the linear model. This option is correct.
• Option D: The sign of the slope \(m\) in the linear model \(y=mx + b\) determines whether the association is positive (\(m>0\)) or negative (\(m < 0\)), not the y - intercept \(b\). So, this option is incorrect.

Answer:

C. The y - intercept gives the initial value at \(x = 0\), and interpreting this value can help determine whether a linear model makes sense.