91Ó°ÊÓ

The article "Air Pollution and Medical Care Use by Older Americans" (Health Affairs [2002]: 207-214) gave data on a measure of pollution (in micrograms of particulate matter per cubic meter of air) and the cost of medical care per person over age 65 for six geographical regions of the United States: \begin{tabular}{lcc} Region & Pollution & Cost of Medical Care \\ \hline North & \(30.0\) & 915 \\ Upper South & \(31.8\) & 891 \\ Decp South & \(32.1\) & 968 \\ West South & \(26.8\) & 972 \\ Big Sky & \(30.4\) & 952 \\ West & \(40.0\) & 899 \\ \hline \end{tabular} a. Construct a scatterplot of the data. Describe any interesting features of the scatterplot. b. Find the equation of the least-squares line describing the relationship between \(y=\) medical cost and \(x=\) pollution. c. Is the slope of the least-squares line positive or negative? Is this consistent with your description of the relationship in Part (a)? d. Do the scatterplot and the equation of the leastsquares line support the researchers' conclusion that elderly people who live in more polluted areas have higher medical costs? Explain.

Step by step solution

01

Draw the Scatterplot

Plot the data points on a graph with 'Pollution' on the x-axis and 'Cost of Medical Care' on the y-axis. Each point corresponds to a particular region.

02

Compute the Least-Squares Line

The equation of a least-squares line is generally \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Use the formula for the slope \( m = \frac{n (\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \) and the formula for the y-intercept \( b = \frac{\sum y - m(\sum x)}{n} \) where \( x \) is pollution, \( y \) is cost, \( n \) is the number of data points, \( \sum xy \) is the sum of the product of \( x \) and \( y \) for all data points, \( \sum x \) and \( \sum y \) are the sums of \( x \) and \( y \) respectively, and \( \sum x^2 \) is the sum of the squares of \( x \).

03

Analyze the slope of the line

Determine whether the slope \( m \) is positive or negative. Analyze whether this is consistent with the correlation observed in the scatterplot in Step 1.

04

Evaluate the Regression Model's Conclusion

Consider whether the equation of the line and the scatterplot support the conclusion that elderly people who live in more polluted areas have higher medical costs. A positive slope would support this, but a high amount of scatter or a low slope might not.

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Key Concepts

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

Scatterplot Analysis

A scatterplot is a powerful visual tool that allows us to observe the relationship between two variables. In this exercise, we plot the data for pollution on the x-axis and the cost of medical care on the y-axis, each point representing a geographical region. By examining the scatterplot:

• We look for patterns or trends, such as whether the points form a line or curve, or if they are scattered randomly.
• We check for any outliers, which are data points that diverge significantly from the trend represented by the rest of the data.
• We also evaluate the direction of any apparent relationship, whether positive (both increase together) or negative (one increases as the other decreases).

In this case, by plotting the pollution and cost of medical care, we can identify whether a trend or correlation exists. This helps in understanding if there's a possible link between pollution levels and medical expenses for older populations across different regions.

Correlation and Causation

Correlation measures the degree to which two variables are related. However, it's crucial to remember that correlation does not imply causation. A strong correlation indicates a relationship, but it doesn’t mean one variable causes the other to change. When analyzing the data:

• We focus on establishing whether there is a correlation between pollution and medical costs, represented by the slope of the regression line.
• A positive correlation and slope would suggest that as pollution increases, so do medical costs, aligning with the initial hypothesis.
• If the slope is negative, it might indicate that higher pollution corresponds to lower medical costs, or there is no direct correlation.

Understanding correlation is key to interpreting results accurately, as other variables might influence both pollution and medical costs, such as regional healthcare policies or economic factors. We need further investigation beyond correlation to establish any causation.

Statistical Interpretation

Statistical interpretation involves making sense of the computed regression line. For this data, the least-squares regression line helps us understand how well pollution levels can predict medical costs:

• The slope of the regression line indicates the expected change in medical costs for a one-unit increase in pollution.
• If the slope is significant and positive, it supports the hypothesis that higher pollution levels lead to increased medical expenses for the elderly.
• The y-intercept offers the predicted medical cost when pollution levels are zero, although it may not always have a practical context.

It's also important to assess the scatter around the regression line:

• A good fit means the points lie close to the line, suggesting a strong predictive relationship.
• Significant scatter or a low correlation coefficient might weaken the confidence in predictions or conclusions drawn.

Through this statistical interpretation, we can evaluate whether the data and the regression model support the researchers' conclusions about pollution and medical costs.