91Ó°ÊÓ

5.56 The article "Organ Transplant Demand Rises Five Times as Fast as Existing Supply" (San Luis Obispo Tribune, February 23,2001 ) included a graph that showed the number of people waiting for organ transplants each year from 1990 to 1999 . The following data are approximate values and were read from the graph in the article: $$ \begin{array}{lc} & \text { Number Waiting } \\ \text { Year } & \begin{array}{c} \text { for Transplant } \\ \text { (in thousands) } \end{array} \\ \hline 1(1990) & 22 \\ 2 & 25 \\ 3 & 29 \\ 4 & 33 \\ 5 & 38 \\ 6 & 44 \\ 7 & 50 \\ 8 & 57 \\ 9 & 64 \\ 10(1999) & 72 \\ \hline \end{array} $$ a. Construct a scatterplot of the data with \(y=\) number waiting for transplant and \(x=\) year. Describe how the number of people waiting for transplants has changed over time from 1990 to 1999 . b. The scatterplot in Part (a) is shaped like segment 2 in Figure \(5.31\). Find a transformation of \(x\) and/or \(y\) that straightens the plot. Construct a scatterplot for your transformed variables. c. Using the transformed variables from Part (b), fit a least-squares line and use it to predict the number waiting for an organ transplant in 2000 (Year 11). d. The prediction made in Part (c) involves prediction for an \(x\) value that is outside the range of the \(x\) values in the sample. What assumption must you be willing to make for this to be reasonable? Do you think this assumption is reasonable in this case? Would your answer be the same if the prediction had been for the year 2010 rather than 2000? Explain.

Step by step solution

01

Construct Scatterplot

Plot data points on a graph with year as the independent variable \(x\) and number waiting for transplant as dependent variable \(y\). Each year value should correspond to the number of people waiting for transplants that year.

02

Describe Trends

Review the scatterplot made. You may notice that the number of people waiting for organ transplants has generally increased over the observed decade.

03

Transform the Scatterplot

Identifying that the scatterplot shows a nonlinear growth trend, we can transform the data using a logarithmic function to straighten the curve into a line. Apply logarithmic function on the \(y\) variables only, creating a new scatterplot with \(x\) = year, \(y' = \ln(y)\) where \(y'\) is the transformed y-variable.

04

Fit a Least-Squares Line and Prediction

Perform a linear regression on the transformed data to fit a least-squares line. The equation of the least-squares line can now be used to predict the number of individuals waiting for organ transplantation in 2000 (Year 11). This could be done by substituting \(x = 11\) into the equation.

05

Discuss Predictive Assumptions

Recognize that the prediction for year 2000 assumes that the past trend will continue into the future. Consequently, it also assumes that the factors influencing the demand for organ transplants over the past years remain the same. Reflect on whether this assumption is reasonable or not. The validity of this prediction would probably decrease for significantly far future years like 2010 since many societal and medical factors can change over a longer period of time.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Organ Transplant Statistics

Understanding organ transplant statistics is crucial when analyzing healthcare data and predicting future trends. In our case, we look at the number of people waiting for organ transplants from 1990 to 1999. Through constructing a scatterplot, we can visually discern the trend of this medical necessity. A clear upward trend over this decade indicates a growing gap between supply and demand in organ transplants. This type of data is essential for health care providers, policymakers, and researchers as it helps them to allocate resources, plan for future needs, and assess the effectiveness of public health initiatives.

It's of particular interest to note the social and technical factors contributing to these trends, such as advancements in medical technology, changes in organ donation policies, or public awareness campaigns. The implications of such statistics are profound, potentially influencing legislation and ethical debates surrounding organ donation and transplantation.

Data Transformation

When we talk about data transformation, we refer to the application of a mathematical modification to the data in order to simplify the underlying relationship for analysis. In the context of organ transplant statistics, the data transformation might be used to straighten a nonlinear relationship in a scatterplot. For instance, applying a logarithmic function—which is a common transformation to address exponential growth—transforms curved data into a linear form that is more amenable to analysis with tools like linear regression.

Data transformation is not only a key technique for simplifying the visualization of trends but also for stabilizing variance, normalizing data, and making it suitable for predictive modeling. It's a fundamental skill for anyone delving into statistical analysis, as it can reveal relationships that might otherwise be obscured in the raw data.

Least-Squares Regression

The method of least-squares regression is a statistical procedure used to find the line that best fits a set of data points. This technique minimizes the sum of the squares of the vertical deviations from each data point to the line. In the organ transplant example, after transforming our data, we use least-squares regression on the transformed variables to create a predictable and understandable relationship between years and the number of patients waiting for transplants.

By finding the equation of the best-fit line, we can make predictions about future values—given that the underlying assumptions hold true. Least-squares regression is widely used in various fields, from economics to engineering, making it an indispensable tool for predicting future outcomes based on historical data.

Extrapolation Assumptions

When we project beyond the range of our original data using a predictive model, we engage in extrapolation. This process is based on the assumption that the patterns observed in the past will continue unchanged into the future. In the scenario of predicting the number of individuals waiting for an organ transplant in 2000, which is beyond our dataset's range, we are relying on this assumption.

However, caution is key. The further we extrapolate, the less reliable our predictions become, as unforeseeable changes can alter the trend. Factors such as policy changes, medical advancements, or demographic shifts may invalidate our assumptions. Therefore, while extrapolation is a valuable tool for short-term forecasts, its assumptions must be carefully considered and questioned for long-term predictions, like for the year 2010, in our organ transplant statistics.