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strojnjashka [21]
3 years ago
5

Let x be the average number of employees in a group health insurance plan, and let y be the average administrative cost as a per

centage 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.

Mathematics
1 answer:
Step2247 [10]3 years ago
6 0

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 <em>x</em> is the average number of employees in a group health insurance plan and <em>y</em> 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 (<em>in this case x</em>) and the dependent variable on the vertical axis (<em>in this case y</em>). 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}

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