Physical fitness testing is an important aspect of athletic...

Physical fitness testing is an important aspect of athletic training. A common measure of the magnitude of cardiovascular fitness is the maximum volume of oxygen uptake during strenuous exercise. A study was conducted on 24 middle-aged men to determine the influence on oxygen uptake of the time required to complete a two-mile run. Oxygen uptake was measured with standard laboratory methods as the subjects performed on a treadmill. The work was published in "Maximal Oxygen Intake Prediction in Young and Middle Aged Males," Journal of Sports Medicine $9,1969,17-22 .$ The data are as follows: $$ \begin{array}{ccc} & y, \text { Maximum } & x, \text { Time } \\ \text { Subject } & \text { Volume of } \mathrm{O}_{2} & \text { in Seconds } \\ \hline 1 & 42.33 & 918 \\ 2 & 53.10 & 805 \\ 3 & 42.08 & 892 \\ 4 & 50.06 & 962 \\ 5 & 42.45 & 968 \\ 6 & 42.46 & 907 \\ 7 & 47.82 & 770 \\ 8 & 49.92 & 743 \\ 9 & 36.23 & 1045 \\ 10 & 49.66 & 810 \\ 11 & 41.49 & 927 \\ 12 & 46.17 & 813 \\ 13 & 46.18 & 858 \\ 14 & 43.21 & 860 \\ 15 & 51.81 & 760 \\ 16 & 53.28 & 747 \\ 17 & 53.29 & 743 \\ 18 & 47.18 & 803 \\ 19 & 56.91 & 683 \\ 20 & 47.80 & 844 \\ 21 & 48.65 & 755 \\ 22 & 53.67 & 700 \\ 23 & 60.62 & 748 \\ 24 & 56.73 & 775 \end{array} $$ (a) Estimate the parameters in a simple linear regression model. (b) Does the time it takes to run two miles have a significant influence on maximum oxygen uptake? Use $H_{0}: \beta_{1}=0$ versus $H_{1}: \beta_{1} \neq 0$ (c) Plot the residuals on a graph against $x$ and comment on the appropriateness of the simple linear model.

Physical fitness testing is an important aspect of athletic training. A common measure of the magnitude of cardiovascular fitness is the maximum volume of oxygen uptake during strenuous exercise. A study was conducted on 24 middle-aged men to determine the influence on oxygen uptake of the time required to complete a two-mile run. Oxygen uptake was measured with standard laboratory methods as the subjects performed on a treadmill. The work was published in "Maximal Oxygen Intake Prediction in Young and Middle Aged Males," Journal of Sports Medicine $9,1969,17-22 .$ The data are as follows:
$$
\begin{array}{ccc}
& y, \text { Maximum } & x, \text { Time } \\
\text { Subject } & \text { Volume of } \mathrm{O}_{2} & \text { in Seconds } \\
\hline 1 & 42.33 & 918 \\
2 & 53.10 & 805 \\
3 & 42.08 & 892 \\
4 & 50.06 & 962 \\
5 & 42.45 & 968 \\
6 & 42.46 & 907 \\
7 & 47.82 & 770 \\
8 & 49.92 & 743 \\
9 & 36.23 & 1045 \\
10 & 49.66 & 810 \\
11 & 41.49 & 927 \\
12 & 46.17 & 813 \\
13 & 46.18 & 858 \\
14 & 43.21 & 860 \\
15 & 51.81 & 760 \\
16 & 53.28 & 747 \\
17 & 53.29 & 743 \\
18 & 47.18 & 803 \\
19 & 56.91 & 683 \\
20 & 47.80 & 844 \\
21 & 48.65 & 755 \\
22 & 53.67 & 700 \\
23 & 60.62 & 748 \\
24 & 56.73 & 775
\end{array}
$$
(a) Estimate the parameters in a simple linear regression model.
(b) Does the time it takes to run two miles have a significant influence on maximum oxygen uptake? Use $H_{0}: \beta_{1}=0$ versus $H_{1}: \beta_{1} \neq 0$
(c) Plot the residuals on a graph against $x$ and comment on the appropriateness of the simple linear model.

Key Concepts

Simple Linear Regression

This concept involves modeling the relationship between two quantitative variables by fitting a linear equation to observed data. In simple linear regression, one variable is treated as the independent variable (predictor) and the other as the dependent variable (response), and the relationship is characterized by an intercept and a slope. This framework is fundamental in statistics for predicting outcomes and understanding linear associations.

Least Squares Estimation

Least squares estimation is the primary method used to determine the best-fitting line in linear regression. The procedure minimizes the sum of the squared differences between the observed responses and the responses predicted by the linear model. This approach provides the estimators for the intercept and slope parameters, which are crucial for making predictions and interpreting the relationship between variables.

Hypothesis Testing for Regression Coefficients

This concept involves evaluating whether the independent variable has a statistically significant effect on the dependent variable. Specifically, it tests if the slope parameter is equal to zero, which would imply no linear relationship. The test is usually carried out using a t-test, where the null hypothesis posits a zero slope, and a significant result indicates that the predictor contributes meaningfully to explaining variation in the response.

Residual Analysis

Residual analysis is the process of examining the differences between observed and predicted values to assess the adequacy of a regression model. It involves plotting residuals to check for patterns that might indicate violations of key assumptions, such as linearity, constant variance, or independence. A well-behaved residual plot should display no systematic structure, supporting the appropriateness of the simple linear model.

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