91Ó°ÊÓ

The following table gives the number of organ transplants performed in the United States each year from 1990 to 1999 (The Organ Procurement and Transplantation Network, 2003): $$ \begin{array}{cc} & \begin{array}{l} \text { Number of } \\ \text { Transplants } \\ \text { Year } \end{array} & \text { (in thousands) } \\ \hline 1(1990) & 15.0 \\ 2 & 15.7 \\ 3 & 16.1 \\ 4 & 17.6 \\ 5 & 18.3 \\ 6 & 19.4 \\ 7 & 20.0 \\ 8 & 20.3 \\ 9 & 21.4 \\ 10 \text { (1999) } & 21.8 \\ \hline \end{array} $$ a. Construct a scatterplot of these data, and then find the equation of the least-squares regression line that describes the relationship between \(y=\) number of transplants performed and \(x=\) year. Describe how the number of transplants performed has changed over time from 1990 to 1999 . b. Compute the 10 residuals, and construct a residual plot. Are there any features of the residual plot that indicate that the relationship between year and number of transplants performed would be better described by a curve rather than a line? Explain.

Step by step solution

01

Constructing the Scatterplot

Use a graphing software or utility to plot each year against the corresponding number of transplants performed. Treat the years as \(x\) values and organ transplants as \(y\) values.

02

Finding Least-Squares Regression Line

Using statistical software or a calculator, input the \(x,y\) pairs and generate the equation of the least-squares regression line, which will have the form \(y=a+bx\). The calculated equation is what describes the relationship between year and number of transplants.

03

Describing Change Over Time

By examining the scatterplot and regression line, describe how the number of transplants performed has changed over time. The sign of the 'b' value will provide insight into whether the change is an increase (positive) or decrease (negative) per year.

04

Calculation of Residuals

Residuals are calculated as the observed \(y\) value minus the predicted \(y\) value for each data point. This can be done using the regression equation obtained in Step 2.

05

Constructing the Residual Plot

The residual plot is constructed by plotting the residuals (obtained from Step 4) against the year. Residuals are plotted on the \(y\)-axis, and the years on the \(x\)-axis.

06

Analyzing Residual Plot

Observing the residual plot will provide insight into whether the relationship between year and number of transplants performed may be better described by a curved relationship. If the residuals exhibit a pattern or trend, a non-linear model may have been a better fit for the data.

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Key Concepts

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

Scatterplot

A scatterplot is a valuable graph that helps you visualize the relationship between two variables. In this context, we use it to graph the number of organ transplants performed against the years, from 1990 to 1999. Each point on the scatterplot represents the number of transplants (the y-axis) for a particular year (the x-axis).
To create a scatterplot:

• Use graphing software or graphing tools to input your data points.
• Place the years on the x-axis and the number of transplants on the y-axis.
• Each pair of values forms a point on the scatterplot.

Once plotted, a scatterplot can show obvious trends or changes over time. For example, you might see that the number of organ transplants tends to increase each year. It's a first step towards linear regression analysis, which helps quantify the relationship detected visually.

Residual Plot

Residual plots are used to evaluate the fit of a regression model. After you've found the least-squares regression line, the next step is to check how well this line predicts your data points. This is done by calculating residuals:

• Residuals are the differences between observed values and the values predicted by your regression line.
• Specifically, they are calculated as the actual number of transplants minus the number of transplants predicted by the regression line.

By plotting the residuals against the x-values (years), you form a residual plot. The purpose of this plot is to identify patterns:

• If the residuals are randomly scattered around zero and show no patterns, the linear model is likely a good fit.
• However, if there is a pattern to the residuals (like a curve), it indicates that the relationship might be better fit by some form of a non-linear model.

So, a residual plot is a simple yet powerful tool for validating your regression analysis.

Time Series Analysis

Time series analysis examines data points collected or recorded at succession of time points. This approach is essential when data display patterns or trends over time. In this example, the data set consists of organ transplant numbers from 1990 to 1999.
It can reveal:

• Trends, indicating general directions in the data over longer periods, like the increase in organ transplants.
• Seasonality, which is regular rise and fall patterns that occur within a specific timeframe, although not relevant in this exercise.

The primary goal of time series analysis here is to identify trends over the years using the scatterplot and the least-squares regression line.
This analysis allows you to make informed predictions and understand underlying data structures. The increase in transplants year-over-year might be analyzed to forecast future trends beyond 1999, helping in making informed future decisions in healthcare and resource allocation.