91Ó°ÊÓ

The accompanying data on \(x=\) treadmill run time to exhaustion (min) and \(y=20-\mathrm{km}\) ski time (min) were taken from the article "Physiological Characteristics and Performance of Top U.S. Biathletes" (Medicine and Science in Sports and Exercise [1995]: \(1302-1310)\) : \(\begin{array}{rrrrrrr}x & 7.7 & 8.4 & 8.7 & 9.0 & 9.6 & 9.6 \\ y & 71.0 & 71.4 & 65.0 & 68.7 & 64.4 & 69.4 \\ x & 10.0 & 10.2 & 10.4 & 11.0 & 11.7 & \\\ y & 63.0 & 64.6 & 66.9 & 62.6 & 61.7 & \end{array}\) $$ \begin{aligned} &\sum x=106.3 \quad \sum x^{2}=1040.95 \\ &\sum y=728.70 \quad \sum x y=7009.91 \quad \sum y^{2}=48390.79 \end{aligned} $$ a. Does a scatterplot suggest that the simple linear regression model is appropriate? b. Determine the equation of the estimated regression line, and draw the line on your scatterplot. c. What is your estimate of the average change in ski time associated with a 1 -min increase in treadmill time? d. What would you predict ski time to be for an individual whose treadmill time is \(10 \mathrm{~min} ?\) e. Should the model be used as a basis for predicting ski time when treadmill time is 15 min? Explain. f. Calculate and interpret the value of \(r^{2}\). g. Calculate and interpret the value of \(s_{e}\).

Step by step solution

01

Create a scatter plot

To decide if a simple linear regression model is appropriate, start by plotting the given \(x\) and \(y\) data on a scatter plot. Examine the pattern of the data. If the points display a linear pattern, then a simple linear regression model might be appropriate.

02

Calculate the regression coefficients

Calculate the slope, \(b_1\), and the intercept, \(b_0\), of the estimated regression line using the formulas: \(b_1 = \frac{(\sum{xy} - n\bar{x}\bar{y})} {(\sum{x^2} - n\bar{x}^2)}\) and \(b_0 = \bar{y} - b_1\bar{x}\) where \(n\) is the number of observations and \(\bar{x}\) and \(\bar{y}\) are the averages of \(x\) and \(y\) respectively.

03

Interpret the slope

The slope of the regression line, \(b_1\), represents the average change in \(y\) (ski time) that would be expected with a 1-minute increase in \(x\) (treadmill time). Use the computed value of \(b_1\) to estimate this average change.

04

Predict the ski time

Plug the treadmill time (\(x = 10\) min) into the equation of the estimated regression line (\(y = b_0 + b_1*x\)) to predict ski time.

05

Discuss the Model’s suitability

Discuss whether the model should be used for predicting ski time when the treadmill time is 15 min. This will be based on the scatter plot and the range of the observed data. If this value of \(x\) falls outside the range of the observed data, extrapolation might not be accurate.

06

Compute \(r^2\)

Compute the coefficient of determination, \(r^2\), using the formula: \(r^2 = \frac{(n\sum xy - \sum x \sum y)^2}{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}\) . The \(r^2\) value explains how much of the variability in \(y\) is accounted for by the model.

07

Calculate \(s_{e}\)

Calculate the standard error of the estimate, \(s_{e}\), using the formula: \(s_{e} = \sqrt{ \frac {1}{n-2} [\sum y^2 - b_0 \sum y - b_1 \sum xy]}\). This measures the average distance that the observed values fall from the regression line.

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Key Concepts

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

Scatterplot Analysis

When examining the relationship between two variables, a scatterplot is an incredibly useful graphical representation. By placing one variable on the x-axis and the other on the y-axis, each individual data point can be plotted, providing a visual insight into any potential correlations.

For instance, in a study analyzing the relationship between treadmill run time and ski time, a scatterplot would allow us to quickly ascertain if there is a discernible pattern. If the data points tend to cluster in a discernible direction, this may indicate a linear relationship, suggesting that a simple linear regression model could be suitable.

A well-constructed scatterplot reveals outliers, tendencies, and the strength of the relationship. It is the first step to determine if further analysis via regression is warranted. The exercise in our example would have one start by plotting each run time against its corresponding ski time and examine the scatter for any linear trend.

Regression Line Equation

The equation of a regression line is at the heart of simple linear regression analysis. It represents the best-fitting straight line through your scatterplot data, calculated by minimizing the distance between the line and the data points.

In mathematical terms, this line can be expressed as:
\[ y = b_0 + b_1x \]
where \( y \) is the dependent variable (ski time in our example), \( x \) is the independent variable (treadmill time), \( b_1 \) is the slope of the line, which tells us how much \( y \) changes for a one-unit increase in \( x \), and \( b_0 \) is the y-intercept, indicating the value of \( y \) when \( x \) equals zero.

The computation of \( b_0 \) and \( b_1 \) involves statistical formulas that leverage the sum of the product of \( x \) and \( y \), the sum of \( x \) squared, and the mean values of both \( x \) and \( y \). Once determined, the equation of the regression line can be used to predict values of \( y \) for given values of \( x \), within the range of the observed data.

Coefficient of Determination

The coefficient of determination, often represented by \( r^2 \), is a key measure in evaluating the fit of our regression model. It essentially quantifies the proportion of the variance in the dependent variable that is predictable from the independent variable.

In the context of our example relating to biathletes' performance, \( r^2 \) would tell us how much of the variance in ski times can be explained by differences in treadmill run times. An \( r^2 \) close to 1 suggests a strong relationship, where much of the variability in ski times can be accounted for by treadmill times. On the other hand, an \( r^2 \) close to 0 indicates a weak relationship.

Interpreting the value of \( r^2 \) helps us understand the explanatory power of the model, crucial for deciding if it's a good predictor for our data. The computation follows a formula involving the number of observations, the sums of the products of \( x \) and \( y \), and the sums of \( x \) and \( y \) squared.

Standard Error of the Estimate

The standard error of the estimate, denoted as \( s_e \), is a measure of the accuracy of predictions made with a regression line. It reflects how closely the data points cluster around the regression line—the smaller the standard error, the closer the points are to the line, indicating more precise predictions.

To put it simpler, \( s_e \) gives us the average distance that the observed values fall from the estimated values on the regression line. It is crucial for assessing the reliability of predictions made by the regression model. In a study on biathletes' performance, for instance, a smaller standard error would imply that the model's predictions of ski times based on treadmill times are more likely to be accurate.

The calculation of \( s_e \) makes use of the residual sums of squares, which represent the differences between observed and predicted values squared and summed, and is adjusted for the number of data points less the number of parameters estimated (minus 2 in simple linear regression).