Question

using equations to represent trend lines
practice describing, drawing, and writing the trend
line of a scatterplot.
does studying improve test scores?
study the scatterplot and trend line. which two points can be used to find the equation of the
trend line?
which points are on the trend line?
(2,45) and (8,90)
(1,30) and (9,95)
(3,50) and (7,65)
(2,30) and (6,70)

Explanation:

Step1: Check each pair of points

For a pair of points \((x_1,y_1)\) and \((x_2,y_2)\), we can check if they lie on the trend - line by visual inspection of the scatter - plot.

• For the pair \((2,45)\) and \((8,90)\):
• Calculate the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{90 - 45}{8 - 2}=\frac{45}{6}=7.5\)
• The equation of the line using the point - slope form \(y - y_1=m(x - x_1)\) (using \((x_1 = 2,y_1 = 45)\)) is \(y-45 = 7.5(x - 2)\), \(y=7.5x+30\)
• When \(x = 2\), \(y=7.5\times2 + 30=15 + 30=45\); when \(x = 8\), \(y=7.5\times8+30=60 + 30=90\)
• By visual inspection of the scatter - plot, these points lie on the trend - line.
• For the pair \((1,30)\) and \((9,95)\):
• Calculate the slope \(m=\frac{95 - 30}{9 - 1}=\frac{65}{8}=8.125\)
• The equation of the line \(y-30 = 8.125(x - 1)\), \(y=8.125x+21.875\)
• When \(x = 1\), \(y = 30\); when \(x=9\), \(y=8.125\times9+21.875=73.125+21.875 = 95\)
• But by visual inspection of the scatter - plot, the point \((1,30)\) is not on the trend - line.
• For the pair \((3,50)\) and \((7,65)\):
• Calculate the slope \(m=\frac{65 - 50}{7 - 3}=\frac{15}{4}=3.75\)
• The equation of the line \(y - 50=3.75(x - 3)\), \(y=3.75x+38.75\)
• When \(x = 3\), \(y=50\); when \(x = 7\), \(y=3.75\times7+38.75=26.25+38.75=65\)
• But by visual inspection of the scatter - plot, the point \((3,50)\) is not on the trend - line.
• For the pair \((2,30)\) and \((6,70)\):
• Calculate the slope \(m=\frac{70 - 30}{6 - 2}=\frac{40}{4}=10\)
• The equation of the line \(y - 30=10(x - 2)\), \(y=10x+10\)
• When \(x = 2\), \(y=30\); when \(x = 6\), \(y=10\times6+10=70\)
• But by visual inspection of the scatter - plot, the point \((2,30)\) is not on the trend - line.

Answer:

\((2,45)\) and \((8,90)\)