Health Insurance: Administrative Cost The following data are...

Health Insurance: Administrative Cost The following data are based on information from Domestic Affairs. 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. $$ \begin{array}{l|rrrrr} \hline x & 3 & 7 & 15 & 35 & 75 \\ \hline y & 40 & 35 & 30 & 25 & 18 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or strong? positive or negative? (c) Use a calculator to verify that $\Sigma x=135, \Sigma x^{2}=7133, \Sigma y=148$, $\Sigma y^{2}=4674$, and $\Sigma x y=3040$. Compute $r$. As $x$ increases from 3 to 75 , does the value of $r$ imply that $y$ should tend to increase or decrease? Explain.

Health Insurance: Administrative Cost The following data are based on information from Domestic Affairs. 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.
$$
\begin{array}{l|rrrrr}
\hline x & 3 & 7 & 15 & 35 & 75 \\
\hline y & 40 & 35 & 30 & 25 & 18 \\
\hline
\end{array}
$$
(a) Make a scatter diagram and draw the line you think best fits the data.
(b) Would you say the correlation is low, moderate, or strong? positive or negative?
(c) Use a calculator to verify that $\Sigma x=135, \Sigma x^{2}=7133, \Sigma y=148$, $\Sigma y^{2}=4674$, and $\Sigma x y=3040$. Compute $r$. As $x$ increases from 3 to 75 , does the value of $r$ imply that $y$ should tend to increase or decrease? Explain.

Key Concepts

Scatter Diagram

A scatter diagram is a graphical tool used to display paired data points on a coordinate plane. It helps in visualizing the relationship between two quantitative variables, allowing one to discern patterns such as trends, clusters, or outliers, and to assess whether a linear relationship might exist.

Correlation Coefficient

The correlation coefficient quantifies the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where values near 1 indicate a strong positive relationship, values near -1 indicate a strong negative relationship, and values near 0 suggest a weak or no linear relationship.

Direction and Strength of Relationship

Understanding the direction (positive or negative) and the strength (strong, moderate, or weak) of a relationship is crucial in data analysis. This concept helps determine whether increases in one variable are associated with increases or decreases in the other, and how consistently this pattern occurs.

Least Squares Regression Line

The least squares regression line is a method for finding the line that best fits the graph of data points by minimizing the sum of the squares of the vertical deviations between the data points and the line. It is widely used for making predictions and summarizing the linear relationship between variables.

Summation Calculations in Statistical Analysis

Summation calculations such as ?x, ?y, ?x^2, ?y^2, and ?xy are foundational in statistics as they are used in formulas to compute the correlation coefficient and determine the parameters of the regression line. These calculations are essential for analyzing and modeling the relationship between variables.

Transcript

00:01 In this video, we'll make a scatter diagram and talk about the fit line of fit and compute the correlation regression.

00:11 So i'm going to take my calculator, stat, edit, and in l1, i've entered the x values, l2, the y values.

00:24 Once you've entered those values in, now we're going to look at a scatter plot.

00:28 So second, y equals, we'll open up the plot screen.

00:33 1, press enter while you're on the word on, select the scatter plot, and then zoom followed by the number 9.

00:44 So one more time real quick, second y equals, enter, make sure it's on scatter plot, which it is, then, and make sure the cursor is on the word on, zoom followed by 9, and there's the scatter plot.

01:07 And again, doing this by hand, not too hard, because this would be 3 ,40.

01:21 Now we need to draw a line that fits the data, drawing it by hand.

01:26 It could look like maybe that.

01:37 If you're drawing it by hand, connecting two points.

01:40 If we're going to use our calculator, do that.

01:43 Then we need to do stat, calc, and we're going to do number four, linear regression.

01:51 Cursor down to calculate.

01:56 And there are the values for our line.

02:00 Now i'm going to copy that real quick.

02:11 So there's the information to help us with our linear aggression model.

02:21 So using this model, i can go ahead and graph this line.

02:26 I can see that my slope is negative.