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Chapter 4: Problem 47

Briefly explain why a large value of \(r^{2}\) is desirable in a regression setting.

Short Answer

Expert verified
A high value of \(r^{2}\) is desirable in a regression setting because it denotes that the model is effectively capturing the variance in the dependent variable. With a large \(r^{2}\), a significant portion of the variation in the output variable is explained by the input variables, indicating a good fit of the data, thus facilitating more accurate predictions.

Step by step solution

01

Understand the meaning of \(r^{2}\)

In the context of a regression model, \(r^{2}\), also known as the coefficient of determination, measures the fraction of the total variation in the dependent variable that is captured by the model. It takes on values between 0 and 1.
02

Interpretation of \(r^{2}\) values

A higher \(r^{2}\) implies that a larger proportion of the variance in the dependent variable is explained by the model. In other words, a higher \(r^{2}\) suggests that the model fits the data better. An \(r^{2}\) of 1 indicates a perfect fit, meaning the model explains all variation in the dependent variable, while an \(r^{2}\) close to 0 indicates that the model explains very little of the variation.
03

Desirability of a high \(r^{2}\)

Given the interpretation of \(r^{2}\), it's clear that a larger \(r^{2}\) is desirable in a regression setting because it signifies that the model is doing a better job of explaining the variation in the dependent variable. A high \(r^{2}\) value can indicate a good fit for the model, making predictions more accurate and the model more useful.

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Most popular questions from this chapter

The paper "School Achievement Strongly Predicts Midlife IQ" (Intelligence [2007]: \(563-567\) ) examined the relationship between school achievement test scores in grades 3 to 8 and IQ measured many years later. The authors reported an association between school achievement test score and midlife IQ, with a correlation coefficient of \(r=0.64\) a. Interpret the value of the correlation coefficient. b. A science writer commenting on this paper (Livescience, April 16,2007\()\) said that the correlation between school achievement test scores and IQ was very strong - even stronger than the correlation between height and weight in adults. Which of the following values for correlation between height and weight in adults is consistent with this statement? Justify your choice. \(\begin{array}{lllll}r=-0.8 & r=-0.3 & r=0.0 & r=0.6 & r=0.8\end{array}\)

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

The paper "Digit Ratio as an Indicator of Numeracy Relative to Literacy in 7-Year-Old British Schoolchildren" (British Journal of Psychology [2008]: \(75-85\) ) investigated a possible relationship between \(x=\) digit ratio (the ratio of the length of the second finger to the length of the fourth finger) and \(y=\) difference between numeracy score and literacy score on a national assessment. (The digit ratio is thought to be inversely related to the level of prenatal testosterone exposure.) The authors concluded that children with smaller digit ratios tended to have larger differences in test scores, meaning that they tended to have a higher numeracy score than literacy score. This conclusion was based on a correlation coefficient of \(r=-0.22 .\) Does the value of the correlation coefficient indicate that there is a strong linear relationship? Explain why or why not.

Draw two scatterplots, one for which \(r=1\) and a second for which \(r=-1\).

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.)
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