Question

given the data points (1,2), (2,3), (3,5), and (4,4), which equation is most likely to represent the line of best fit?
○ y = x + 1.5
○ y = x - 1
○ y = 2x
○ y = 0.5x + 2

Explanation:

Step1: Calculate mean of x values

$\bar{x} = \frac{1+2+3+4}{4} = \frac{10}{4} = 2.5$

Step2: Calculate mean of y values

$\bar{y} = \frac{2+3+5+4}{4} = \frac{14}{4} = 3.5$

Step3: Verify line passes through $(\bar{x},\bar{y})$

Check each option by substituting $x=2.5$:

• For $y=x+1.5$: $y=2.5+1.5=4

eq 3.5$

• For $y=x-1$: $y=2.5-1=1.5

eq 3.5$

• For $y=2x$: $y=2\times2.5=5

eq 3.5$

• For $y=0.5x+2$: $y=0.5\times2.5+2=1.25+2=3.25$ (closest to 3.5, and we can confirm by checking sum of squared errors)

Step4: Calculate sum of squared errors for valid option

For $y=0.5x+2$:

• $(1,2)$: $(2 - (0.5+2))^2=(2-2.5)^2=0.25$
• $(2,3)$: $(3 - (1+2))^2=(3-3)^2=0$
• $(3,5)$: $(5 - (1.5+2))^2=(5-3.5)^2=2.25$
• $(4,4)$: $(4 - (2+2))^2=(4-4)^2=0$

Total error: $0.25+0+2.25+0=2.5$
For other options, total error is larger (e.g., $y=x+1.5$ has total error $(2-2.5)^2+(3-3.5)^2+(5-4.5)^2+(4-5.5)^2=0.25+0.25+0.25+2.25=3$)

Answer:

D. $y=0.5x+2$