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Chapter 4: Problem 40

Briefly explain why a small value of \(s_{e}\) is desirable in a regression setting.

Short Answer

Expert verified
A small \(s_{e}\), or standard error of estimate, is desirable in a regression setting because it suggests a high level of precision in the model's predictions, with a minimal level of unexplained variability in the dependent variable. It indicates that the observed data points closely fit the estimated regression line, implying that the model is reliable and accurate.

Step by step solution

01

Understand the Role of \(s_{e}\) in Regression

In regression analysis, the standard error of the estimate, or \(s_{e}\), indicates how close the data are to the fitted regression line. It's the standard deviation of the residuals.
02

Explain the Desirability of Small \(s_{e}\)

A small \(s_{e}\) is desirable as it indicates that the observed values or data points are closely packed around the estimated regression line. This means that there's a high degree of precision in the predictions made by the model, and a minimal level of unaccounted variability in the response variable, hence maximizing the explanatory power of the independent variables.
03

Conclude

Overall, a small \(s_{e}\) indicates that the regression model predicts the dependent variable well, with very little inaccuracy or variability. Therefore, in a regression setting, one would ideally want the \(s_{e}\) to be as small as possible to ensure the accuracy and reliability of the model's predictions.

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Most popular questions from this chapter

It may seem odd, but biologists can tell how old a lobster is by measuring the concentration of pigment in the lobster's eye. The authors of the paper "Neurolipofuscin Is a Measure of Age in Panulirus argus, the Caribbean Spiny Lobster, in Florida" (Biological Bulletin [2007]: 55-66) wondered if it was sufficient to measure the pigment in just one eye, which would be the case if there is a strong relationship between the concentration in the right eye and the concentration in the left eye. Pigment concentration (as a percentage of tissue sample) was measured in both eyes for 39 lobsters, resulting in the following summary quantities (based on data from a graph in the paper): $$ \begin{array}{cll} n=39 & \sum_{x}=88.8 & \sum y=86.1 \\ \sum x y=281.1 & \sum x^{2}=288.0 & \sum y^{2}=286.6 \end{array} $$ An alternative formula for calculating the correlation coefficient that doesn't involve calculating the z-scores is $$ r=\frac{\sum_{x y}-\frac{\left(\sum x\right)\left(\sum y\right)}{n}}{\sqrt{\sum x^{2}-\frac{\left(\sum x\right)^{2}}{n}} \sqrt{\sum y^{2}-\frac{\left(\sum y\right)^{2}}{n}}} $$ Use this formula to calculate the value of the correlation coefficient, and interpret this value.

For a given data set, the sum of squared deviations from the line \(y=40+6 x\) is \(529.5 .\) For this same data set, which of the following could be the sum of squared deviations from the least squares regression line? Explain your choice. i. 308.6 ii. 529.6 iii. 617.4

Briefly explain why it is important to consider the value of \(s\) in addition to the value of \(r^{2}\) when evaluating the usefulness of the least squares regression line.

Data on \(y=\) time to complete a task (in minutes) and \(x=\) number of hours of sleep on previous night were used to find the least squares regression line. The equation of the line was \(\hat{y}=12-0.36 x .\) For this data set, would the sum of squared deviations from the line \(y=12.5-0.5 x\) be larger or smaller than the sum of squared deviations from the least squares regression line? Explain your choice. (Hint: Think about the definition of the least- squares regression line.)

A sample of automobiles traveling on a particular segment of a highway is selected. Each one travels at roughly a constant rate of speed, although speed does vary from auto to auto. Let \(x=\) speed and \(y=\) time needed to travel this segment. Would the sample correlation coefficient be closest to \(0.9,0.3,-0.3,\) or \(-0.9 ?\) Explain.
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