Question

10
the data to the left shows 9 students gpa and scholarship
gpa scholarship money dollars awarded.
3.1 $15,000
4.2 $85,000
2.1 $0
3.4 $24,000
3.8 $29,000
2.7 $10,000
2.4 $0
3.9 $34,000
4.5 $100,000
which of the following equations best represents the equation
for the line of best fit?
$y = 38,000x - 96,000$
$y = 38,000x + 96,000$
$y = 38,000x$
$y = 40,000x + 100,000$

Explanation:

Step1: Analyze the general form of a linear equation

The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

Step2: Estimate the slope

If we consider two points \((x_1,y_1)\) and \((x_2,y_2)\). Let's take \((x_1 = 3.1,y_1=15000)\) and \((x_2 = 4.2,y_2 = 85000)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{85000 - 15000}{4.2-3.1}=\frac{70000}{1.1}\approx63636\) (this is a rough check, but another way is to note that when \(x = 3.1\), if \(y=38000x-96000\), then \(y=38000\times3.1-96000=117800 - 96000=21800\) (close to \(15000\)), if \(x = 4.2\), \(y=38000\times4.2-96000=159600 - 96000 = 63600\) (close to \(85000\) considering data spread). For \(y = 38000x+96000\), when \(x = 3.1\), \(y=38000\times3.1 + 96000=117800+96000=213800\) (too large). For \(y = 38000x\), when \(x = 3.1\), \(y=38000\times3.1=117800\) (too large). For \(y=40000x + 100000\), when \(x = 3.1\), \(y=40000\times3.1+100000=124000 + 100000=224000\) (too large).

Answer:

\(y = 38000x-96000\)