Question

This table shows the SAT scores and final college GPA of ten students. SAT and GPA Correlation SAT Score College GPA 650 2.43 986 3.0 860 2.87 1250 3.8 1400 3.74 1440 3.62 622 2.3 1050 3.18 1170 3.3 1560 3.92 Which graph shows the line of best fit for this data?

Answer 1

The first graph is the correct answer. Its the graph that has a line running through it that's starts at the right angle.

Answer 2

The graph is below.

Explanation:

Linear regression is a linear approach to modelling the relationship between a dependent variable and one or more independent variables.

Let X be the independent variable and Y be the dependent variable. We will define a linear relationship between these two variables as follows:

`Y=bX+a`

We have the following data:

`\left\begin{array}{cc}\mathrm{SAT \:Score}&\mathrm{College GPA}\\650&2.43\\986&3.0\\860&2.87\\1250&3.8\\1400&3.74\\1440&3.62\\622&2.3\\1050&3.18\\1170&3.3\\1560&3.92\end{array}\right \\`

To find the line of best fit for the points, follow these steps:

Step 1: Find
`X\cdot Y` and
`X\cdot X` as it was done in the table.

Step 2: Find the sum of every column:

`\sum{X} = 10988 ~,~ \sum{Y} = 32.16 ~,~ \sum{X \cdot Y} = 36950.3 ~,~ \sum{X^2} = 13022280`

Step 3: Use the following equations to find intercept a and slope b:

`\begin{aligned}        a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} =             ( 32.16 \cdot 13022280 - 10988 \cdot 36950.3)/( 10 \cdot 13022280 - 10988^2) \approx 1.348 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2}        = ( 10 \cdot 36950.3 - 10988 \cdot 32.16 )/( 10 \cdot 13022280 - \left( 10988 \right)^2) \approx 0.002\end{aligned}`

Step 4: Assemble the equation of a line

`\begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~1.348 ~+~ 0.002 \cdot x\end{aligned}`

The graph of the regression line is: