Question

Let x be the average number of employees in a group health insurance plan, and let y be the average administrative cost as a percentage of claims.
x 3 7 15 32 74
y 40 35 30 25 17

a) Make a scatter diagram of the data and visualize the line you think best fits the data.

Answer 1

A scatter diagram has points that show the relationship between two sets of data.

We have the following data,

$\left\begin{array}{ccccccc}\mathrm{x}&3&7&15&32&74\\\mathrm{y}&40&35&30&25&17\end{array}\right$

where x is the average number of employees in a group health insurance plan and y is the average administrative cost as a percentage of claims.

To make a scatter diagram you must, draw a graph with the independent variable on the horizontal axis (in this case x) and the dependent variable on the vertical axis (in this case y). For each pair of data, put a dot or a symbol where the x-axis value intersects the y-axis value.

Linear regression is a way to describe a relationship between two variables through an equation of a straight line, called line of best fit, that most closely models this relationship.

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 below table.

Step 2: Find the sum of every column:

$\sum{X} = 131 ~,~ \sum{Y} = 147 ~,~ \sum{X \cdot Y} = 2873 ~,~ \sum{X^2} = 6783$

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

$\begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 147 \cdot 6783 - 131 \cdot 2873}{ 5 \cdot 6783 - 131^2} \approx 37.05 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 5 \cdot 2873 - 131 \cdot 147 }{ 5 \cdot 6783 - \left( 131 \right)^2} \approx -0.292\end{aligned}$

Step 4: Assemble the equation of a line

$\begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~37.05 ~-~ 0.292 \cdot x\end{aligned}$