91Ó°ÊÓ

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 \(\Sigma x=135, \Sigma x^{2}=7133, \quad \Sigma y=148\), \(\Sigma y^{2}=4674\), 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.

Step by step solution

01

Create the Scatter Plot

Begin by plotting the given pairs of data (3,40), (7,35), (15,30), (35,25), (75,18) on a graph with the x-axis representing the number of employees, x, and the y-axis representing the administrative cost percentage, y. These points should form the scatter plot from which we can visualize the relationship between the variables.

02

Draw the Line of Best Fit

Visually inspect the scatter plot and attempt to draw a line that best follows the trend of the data points. The line should have roughly equal numbers of points above and below it, attempting to minimize the distances between the points and the line.

03

Determine Correlation Characteristics

Observe the scatter plot and the line of best fit to assess the correlation. If the line is generally sloping downward from left to right, the correlation is negative. If data points are tightly clustered around the line, the correlation is strong; if more spread out, moderate; and if very scattered, low.

04

Calculate the Correlation Coefficient, r

Use the formula \[r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}\]to calculate the correlation coefficient, where n is the number of data points \(n = 5\). Substitute given values \(\Sigma x = 135, \Sigma y = 148, \Sigma xy = 3040, \Sigma x^2 = 7133, \Sigma y^2 = 4674\).Calculate to find \(r\approx -0.96\), indicating a strong negative correlation.

05

Interpret the Correlation Coefficient

Since \(r \approx -0.96\), which is very close to -1, it implies a strong negative correlation. This suggests that as the number of employees \(x\) increases, the administrative cost percentage \(y\) tends to decrease.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Plot

A scatter plot is a type of graph that is used to represent the relationship between two variables. It is helpful for visualizing patterns, trends, and potential correlations between these variables. To create a scatter plot, we place one variable along the x-axis and the other variable on the y-axis. Each point on the plot thus represents a pair of values.
In the context of the exercise, we plotted the average number of employees in a group health insurance plan on the x-axis and the average administrative cost as a percentage of claims on the y-axis.
This initial visual representation allows us to quickly assess whether there is a pattern or trend between the variables, such as seeing whether costs increase, decrease, or stay the same as the number of employees increases.

Line of Best Fit

The line of best fit is a straight line drawn through the center of a group of data points on a scatter plot. Its purpose is to show the overall direction (trend) of the data. When drawing the line of best fit, we try to make sure that the line is as close as possible to all the data points, with roughly equal scattering of points above and below the line.
A well-placed line of best fit helps in identifying the correlation between the two variables. In our example, once we plotted the data on a scatter plot, we tried to fit a line that showed the decreasing trend of administrative costs % as the number of employees increased. This visual helps solidify whether the correlation is positive or negative and how strong it is.

Negative Correlation

Negative correlation refers to a relationship between two variables in which one variable increases, while the other decreases. On a scatter plot, a negative correlation will often result in a line of best fit that slopes downwards from left to right.
In the provided exercise, we calculated a correlation coefficient of approximately \(r \approx -0.96\), which indicates a strong negative correlation between the average number of employees and the administrative cost percentage. This suggests that as the average number of employees in a plan increases, the administrative costs tend to decrease.
Identifying a negative correlation is important in statistical analysis as it helps us understand underlying relationships and informs decisions based on the expected behavior of variables.

Statistical Analysis

Statistical analysis involves applying statistical methods to collected data in order to interpret it and draw conclusions. It encompasses a variety of methods including describing data, calculating probabilities, and developing predictive models.

• In our context, we used statistical analysis to calculate the correlation coefficient, \(r\), using the given sums and formulas.
• It helps us quantify the strength and direction of the relationship between the number of employees and administrative cost percentage.

With the correlation coefficient \(r \approx -0.96\), our analysis revealed a strong negative correlation. Such analysis is essential for making informed decisions and predictions. For example, insurance companies might use this data to optimize their costs and set premiums.