Question

the scatterplot shows the federal minimum wage rates in the united states, calculated every five years from 1970 (year 1 on the graph) to 2005 (year 8). which is the best equation of the trend line for the scatterplot? y = - 0.45x + 1.55 y = 0.45x - 1.55 y = 0.45x + 1.55 y = - 0.45x - 1.55

Explanation:

Step1: Analyze the slope sign

Looking at the scatter - plot, as \(x\) (years) increases, \(y\) (minimum wage) also increases. So the slope \(m\) of the trend line should be positive.

Step2: Use the slope - intercept form \(y = mx + b\)

The general form of a linear equation is \(y=mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept.
We know that when \(x = 1\) (1970), \(y\approx1.5\).
If we substitute \(x = 1\) into the equation \(y=0.45x + 1.55\), we get \(y=0.45\times1+1.55=0.45 + 1.55=2\) (a reasonable approximation considering the scatter - plot data points).
If we consider the \(y\) - intercept: when \(x = 0\) (a year before 1970), the \(y\) - value should be positive. For \(y = 0.45x+1.55\), when \(x = 0\), \(y = 1.55\) which is reasonable.
For \(y=-0.45x + 1.55\), the slope is negative (contradicts the upward trend of the data). For \(y = 0.45x-1.55\), when \(x = 1\), \(y=0.45\times1-1.55=-1.1\) (negative minimum - wage which is not possible in this context). For \(y=-0.45x - 1.55\), the slope is negative (contradicts the upward trend) and the \(y\) - intercept is negative (not possible for minimum - wage data).

Answer:

\(y = 0.45x+1.55\)