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The value of $s$ for a relationship between the number of hours and the number of gallons of water remaining in a tank is calculated using computer software. Identify and interpret the value of $s$ by completing the sentence.




Predictor		CoefSE CoefTP

Constant		107.073.7428.620.000

Number of Hours		-7.5710.741-10.220.000

$S =4.80079$		\multicolumn{2}{|l|}{$R - Sq =94.578$}\multicolumn{2}{|l|}{R-Sq $( adj )=93.66 \%$}






The value of $s = \text{□}$

$\text{□}$ tells us that the average residual is about $\text{□}$ gallons.

Verified Answer :

  • The value of \(s\) is found in the provided table as \(s = 4.80079\).
  • This value represents the standard deviation of the residuals in the regression analysis.
  • It signifies that the typical difference between the observed and predicted number of gallons is approximately 4.80079 gallons.
  • The completed sentences are: The value of \(s = \boxed{4.80079}\). \(\boxed{4.80079}\) tells us that the average residual is about \(\boxed{4.80079}\) gallons.

Explanation

  1. Identify the value of s from the table

In regression analysis, the standard error of the estimate, denoted as \(s\) (or sometimes \(s_e\)), is a crucial statistic that summarizes the dispersion of the observed data points around the regression line. It provides a measure of the accuracy of the predictions made by the model. By looking at the provided regression output table, we can easily identify this value. In the row containing \(S = 4.80079\), we find the value of \(s\). Therefore, \(s = 4.80079\).

  1. Interpret the meaning of s in context

The value \(s = 4.80079\) represents the standard deviation of the residuals. In the context of our regression model, the residuals are the differences between the actual observed number of gallons of water remaining in the tank and the values predicted by our linear equation. Thus, the value \(s = 4.80079\) indicates that the observed number of gallons typically deviates from the regression line by approximately 4.80079 gallons. This value helps us quantify the typical vertical distance of the data points from the model's line, providing an estimate of the "average error" in our predictions.

Examples

This concept is widely used in data science and engineering to evaluate how well a linear model fits real-world data. For example, if you are modeling battery discharge, the standard error \(s\) tells you how much the actual battery life might vary from the model's prediction, helping you set expectations for real-world performance. Calculating \(s\) involves taking the square root of the sum of squared residuals divided by the degrees of freedom: \(s = \sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n - 2}}\).

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