Question

Liver Transplants The table shows the numbers \(L\) of liver transplant procedures performed in the United States in the years 1995 to \(2005 .\) (Source: U.S. Department of Health and Human Services) $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 1995 & 1996 & 1997 & 1998 \\ \hline \text { Transplants, } L & 3818 & 3918 & 4005 & 4356 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|} \hline \text { Year } & 1999 & 2000 & 2001 & 2002 \\ \hline \text { Transplants, } L & 4586 & 4816 & 5177 & 5326 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline \text { Year } & 2003 & 2004 & 2005 \\ \hline \text { Transplants, } L & 5671 & 6168 & 6444 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=5\) corresponding to \(1995 .\) (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model from part (b) in the same viewing window as the scatter plot of the data. (d) Use the graph of the model from part (c) to predict the number of liver transplant procedures performed in \(2008 .\)

Short answer

The regression model will yield a quadratic equation in the form \(y = ax^2 + bx + c\) which provides a mathematical model for the liver transplants per year. The forecast for the year 2008 based on this model can be calculated by substituting \(t = 18\) into the equation.

Step-by-step solution

Create a scatter plot of the data

Represent the years as 't' starting from 5 (1995). Hence, create a list of x-coordinate values as follows: [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Create a list of y-coordinates as per the transplant data: [3818, 3918, 4005, 4356, 4586, 4816, 5177, 5326, 5671, 6168, 6444]. Use these data to plot a scatter plot using a graphing utility.

Find a quadratic regression model for data

With the x-values and y-values input in the graphing utility, utilize its regression feature to find the quadratic function that best fits the data points. The utility provides output in the form of \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants determined by the regression model.

Graph the regression model and the scatter plot

Plot the quadratic function from step 2 on the same graph as the scatter plot. This visualization will show how closely the fitted quadratic equation corresponds to the actual data points.

Predicting future values with the regression model

To predict the number of liver transplant procedures performed in the year 2008, substitute \(t = 18\) (as 1995 corresponds to 5 and 2008 corresponds to 18) into the quadratic equation derived from the regression model. The computed function value will be the prediction.

Key concepts

Scatter Plot

A scatter plot is a type of graph used to represent data points on a two-dimensional plane. This method is particularly helpful in identifying the relationship or pattern between two variables. For instance, in the context of predicting liver transplants, a scatter plot can be created using the year as the x-axis and the number of liver transplants as the y-axis. Each point on the scatter plot represents a particular year's data, showing how the number of transplants changes over time.
A scatter plot can visually show clusters, trends, and even possible outliers which deviate from the expected range.

• Clusters indicate strong groupings of data points, potentially showing a stable trend over time.
• An upward trend might suggest that as the years increase, so does the number of transplants.

By plotting your data this way, you gain a clearer insight into underlying patterns, which set the stage for more advanced analyses like data modeling.

Data Modeling

Data modeling involves creating a mathematical representation or model of a dataset to understand the dynamics of a given situation. Using a scatter plot is the first step to understanding the distribution and pattern of data. But to predict or understand future trends, a model is necessary.
The model usually comes from a regression analysis, which can be linear, quadratic, or another type of polynomial fitting depending on the data's complexity. When using quadratic regression, the aim is to define an equation of the form \( y = ax^2 + bx + c \). This equation should minimize the difference between the predicted and observed values.

• This fitting is achieved with techniques such as least squares.
• The coefficients \( a \), \( b \), and \( c \) are calculated to generate the best-fit curve that captures the essence of the data relationship.

Such models are crucial as they provide a systematic approach to understand trends and make informed predictions based on historical data.

Graphing Utility

A graphing utility is a tool, either software or calculator-based, used to create representations of data and models for easy analysis. These utilities make complex computations and graphing tasks accessible and manageable.
In the context of liver transplant data, a graphing utility helps in two primary ways:

• Creating scatter plots, giving a visual sense of how data is distributed.
• Executing regressions and graphing resultant models to compare them with actual data points.

With the built-in capabilities of such utilities, one can quickly input data points, apply mathematical functions, and visualize results in real-time. This visualization is instrumental, as seeing the modeled equation plotted alongside real data underscores the accuracy and nuances of the data model selected.
Graphing utilities simplify the trial-and-error process of finding the most suitable mathematical model for a given problem.

Predictive Analysis

Predictive analysis uses historical data to predict future outcomes. With liver transplant data, once a suitable model has been established through data modeling, predictive analysis allows calculation and estimation beyond the available data points.
The quadratic regression model, for instance, provides a basis to forecast future values. By substituting a future year value into the quadratic equation, one can predict likely outcomes. In this scenario, predicting the number of liver transplants in 2008 involves using the model to substitute for \( t = 18 \).

• This mathematical approach offers a quantified forecast, shaping potential decisions based on future insights.
• The reliability of predictions improves with the accuracy of the underlying model.

Predictive analysis is widely applied across fields like healthcare, finance, and engineering, illustrating its versatility in foreseeing trends and preparing for future requirements.