91Ó°ÊÓ

The data in the accompanying table is from the paper "Six-Minute Walk Test in Children and Adolescents" (The journal of Pediatrics [2007]: 395-399). Two hundred and eighty boys completed a test that measures the distance that the subject can walk on a flat, hard surface in 6 minutes. For each age group shown in the table, the median distance walked by the boys in that age group is also given. \begin{tabular}{ccc} & Representative Age (Midpoint of Age Group) & Median Six-minute Walk Distance \\\ Age Group & 4 & (meters) \\ \hline \(3-5\) & 7 & \(544.3\) \\ \(6-8\) & 7 & \(584.0\) \\ \(9-11\) & 10 & \(667.3\) \\ \(12-15\) & \(13.5\) & \(701.1\) \\ \(16-18\) & 17 & \(727.6\) \\ \hline \end{tabular} a. With \(x=\) representative age and \(y=\) median distance walked in 6 minutes, construct a scatterplot. Does the pattern in the scatterplot look linear? b. Find the equation of the least-squares regression line that describes the relationship between median distance walked in 6 minutes and representative age. c. Compute the five residuals and construct a residual plot. Are there any unusual features in the plot?

Step by step solution

01

Constructing the Scatterplot

The scatterplot will aid in the visualization of data points. Each data point will correspond to one age group's representative age (\(x\)) and their median six-minute walk distance (\(y\)). By plotting all the points, we can visually deduce whether the data pattern appears linear.

02

Computing the Least-Squares Regression Line

Compute the least-squares regression line equation, this is expressed as \(y = ax + b\), where \(a\) represents the slope and \(b\) the intercept. The slope \(a\) can be calculated with the formula \(a = r * (sy/sx)\), where \(r\) is the correlation coefficient, \(sy\) is the standard deviation of the \(y\) variable, and \(sx\) is the standard deviation of the \(x\) variable. The intercept \(b\) is then calculated with the formula \(b = \overline{y} - a*\overline{x}\), where \(\overline{y}\) and \(\overline{x}\) are the means of \(y\) and \(x\) respectively.

03

Computing and Interpreting Residuals

Each residual corresponds to the difference between an observed value of \(y\) and the corresponding fitted value of \(y\). These can be calculated by subtracting the observed \(y\) value from the predicted \(y\) value for each data point using the regression line. Once all residuals are computed, they should be plotted against the independent variable \(x\). Observing any visible patterns in the residuals can help identify potential deviations from the linear regression model. Specifically, if residuals seem to increase or decrease systematically with \(x\), it may suggest a nonlinear relationship that isn’t captured by our model.

04

Evaluating the Residual Plot for Unusual Features

After constructing the residual plot, look for any clear patterns, such as trends or curvature. If the residuals appear to be randomly dispersed around the horizontal axis, it can be inferred that a linear regression model is appropriate for the data. Observable trends, curvature, or other patterns in the residuals suggest that a linear regression model may not be the best fit.

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Key Concepts

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

Scatterplot Construction

When studying how two quantitative variables relate to each other, a scatterplot is an indispensable tool. It enables us to visualize the relationship by plotting points that each represent a pair of values for the two variables. In constructing a scatterplot, one variable is designated as the independent variable, usually plotted on the x-axis, and the other as the dependent variable, plotted on the y-axis.

To create a clear and informative scatterplot, follow these steps:

• Select a scale for each axis that reasonably uses the grid and allows for all data points to be plotted without crowding.
• Label each axis with the variable name and the units of measurement.
• Plot each pair of values as a point on the graph where the two values intersect.
• Review the scatterplot to ensure that all points have been accurately plotted.

For our exercise, we see that as the representative age increases, there seems to be an increase in the median distance walked, suggesting a positive linear relationship. However, to confirm this, we must analyze the scatterplot for any clear patterns or deviations from linearity.

Least-Squares Regression Line

The least-squares regression line is a statistically derived line that best fits the data in a scatterplot, thereby allowing us to predict the dependent variable based on the independent variable. The mathematical formulation for the least-squares regression line is given by the equation \( y = ax + b \) where \(a\) is the slope of the line and \(b\) is the y-intercept.

The slope, \(a\), is calculated as the product of the correlation coefficient, \(r\), and the ratio of the standard deviations of the dependent and independent variables, \(sy/sx\). The y-intercept, \(b\), is found by subtracting the product of the slope and the mean of the independent variable, \(a\cdot\overline{x}\), from the mean of the dependent variable, \(\overline{y}\).

Having the regression line enables us to predict the median distance walked for any given age not present in the data. The resulting equation is a powerful tool but must be used judiciously. It's essential to remember that this line is based on the given dataset and the relationship it represents may not hold outside the data's range or in a different context.

Residual Analysis

Residual analysis allows us to assess the fit of the least-squares regression line to our data. A 'residual' is the difference between the observed value and the value predicted by our regression line, essentially measuring the error in the prediction for each data point.

To conduct a residual analysis:

• Calculate each residual by subtracting the predicted value of the dependent variable from the observed value.
• Plot the residuals on a graph with the independent variable on the x-axis and the residuals on the y-axis.
• Examine the residual plot for patterns, such as increasing or decreasing trends, curvature, or clustering, that may indicate a non-linear relationship.

For the data on the walking distance of boys, we compute the residuals to check whether the variance is consistent across all ages or if there are any outliers. A well-fitting linear model will have residuals that are roughly symmetrically distributed around zero, with no evident patterns. Any systematic structure in the residuals plot could signal a need to reconsider our model—for example, by looking into a non-linear relationship or investigating variables that might be affecting the residuals.