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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: $$ \begin{array}{lcc} \text { Region } & \text { Pollution } & \text { Cost of Medical Care } \\ \hline \text { North } & 30.0 & 915 \\ \text { Upper South } & 31.8 & 891 \\ \text { Deep South } & 32.1 & 968 \\ \text { West South } & 26.8 & 972 \\ \text { Big Sky } & 30.4 & 952 \\ \text { West } & 40.0 & 899 \\ & & \\ \hline \end{array} $$ 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 least-squares line support the researchers' conclusion that elderly people who live in more polluted areas have higher medical costs? Explain.

Step by step solution

01

Construct the scatterplot

To construct a scatterplot of the data, put the pollution measure on the x-axis and the cost of medical care on the y-axis. Then plot each region's point as (pollution, cost of medical care). Analysis of the scatterplot will reveal the trend - does the cost generally increase or decrease as pollution increases?

02

Calculate the least-squares line equation

The least squares line can be calculated using the formula \(y = a + bx\), where \(b = r(s_y/s_x)\) and \(a= \bar{y} - b\bar{x}\). In these formulas, \(\bar{y}\) and \(\bar{x}\) are the means of the y and x data respectively, \(s_y\) and \(s_x\) are the standard deviations of the y and x data, and \(r\) is the correlation coefficient between the y and x data.

03

Determine the slope of the least-squares line

The slope of the least squares line, \(b\), can be determined from the calculation in the previous step. If \(b > 0\), the slope is positive, meaning the cost of medical care increases as pollution increases. If \(b < 0\), the slope is negative, meaning the cost of medical care decreases as pollution increases.

04

Connect the scatterplot and equation to the researchers' conclusion

Based on the scatterplot and the equation of the least squares line, determine if there is evidence to support the researchers' conclusion. If the slope is positive and the scatterplot shows a general trend of higher costs with higher pollution levels, this would support the researchers' conclusion.

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Key Concepts

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

Least-Squares Line

The concept of the Least-Squares Line is central to understanding relationships between two variables in statistics. In this context, it helps us explore how pollution levels may affect medical costs for older adults. The least-squares line is a straight line that minimizes the sum of the squared differences between the observed points and the line itself, allowing us to summarize the overall pattern in the data.

For this dataset, we compute the least-squares line equation using the formula:

• \( y = a + bx \)

where:

• \( b = r \frac{s_y}{s_x} \)
• \( a = \bar{y} - b\bar{x} \)

Here, \( b \) represents the slope of the line, and \( a \) is the intercept. The x-variable is pollution, and the y-variable is the medical cost.

The slope \( b \) indicates the rate of change in the y-values with each unit of change in x-values.

Correlation Coefficient

The Correlation Coefficient, often represented as \(r\), is a statistical measure that expresses the extent to which two variables are linearly related. In simple terms, it tells us how strongly the pollution levels and medical costs are linked together in this study.

• A positive \(r\) indicates that as pollution increases, medical costs tend to also increase.
• A negative \(r\) suggests that as pollution levels rise, the medical costs might decrease.
• An \(r\) value close to 0 implies little to no linear relationship.

For this data, finding \(r\) involves using the means and standard deviations of the datasets. Calculating \(r\) correctly is crucial because it influences the steepness and direction of the least-squares line. This correlation helps researchers understand whether there is a meaningful trend to discuss in medical policy or environmental health guidance.

Data Interpretation

Interpreting the scatterplot data and the calculated equations is crucial for making informed conclusions. From the scatterplot, each point represents pollution levels against medical costs for a particular region.

Important aspects of interpretation include:

• Identifying trends: Does the scatterplot show an increasing pattern, implying higher medical costs with more pollution?
• Assessing outliers or anomalies: Are there regions significantly deviating from the pattern?
• Consistency: Does the least-squares line capture the main trend observed in the scatterplot?

Upon analysis, if the line slopes upwards and aligns with the general pattern of the points, it supports the conclusion about pollution leading to increased healthcare costs.

Pollution and Health Costs

The relationship between pollution and health costs is a significant concern in public health research, and understanding their interplay informs policy decisions. Increased pollution can exacerbate health issues, particularly among vulnerable populations like older adults. This can lead to a rise in healthcare costs, as more medical care becomes necessary to address pollution-related health problems.

The study highlighted provides a practical example of how data about pollution and healthcare costs can be quantified and analyzed:

• By constructing a statistical analysis like the scatterplot and least-squares line, we visually and numerically interpret the impact of environmental factors on health expenses.
• This understanding helps in framing policies to minimize pollution-related health risks and healthcare expenses.

This alignment between data interpretation and real-world implications highlights the importance of informed environmental and health policy.