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Chapter 5: Problem 27

Explain why the slope \(b\) of the least-squares line always has the same sign (positive or negative) as does the sample correlation coefficient \(r\).

Short Answer

Expert verified
The sign of the slope \(b\) of the least-squares line and the sample correlation coefficient \(r\) always coincide because the slope formula \(b = r \cdot \frac{s_y}{s_x}\) includes \(r\) directly. Given that standard deviations are always positive, the sign of the slope is solely determined by the sign of the correlation coefficient.

Step by step solution

01

Understanding the variables

Based on the formula for the slope of the least squares regression line \(b = r \cdot \frac{s_y}{s_x}\), where \(r\) represents the sample correlation coefficient and \(s_y, s_x\) are the standard deviations of \(y\) and \(x\) respectively. These standard deviations are always positive.
02

Understanding the relationship between the variables

It's worth noting that the sign of the slope \(b\) solely relies on the sample correlation coefficient \(r\), as standard deviations \(s_y, s_x\) will always be positive by definition. The sign of \(r\) determines the sign of \(b\).
03

Conclusion about the relationship of the signs

Given the relationship in step 2, we can therefore conclude that the sign (positive or negative) of the slope \(b\) of the least-squares line will always coincide with the sign of the sample correlation coefficient \(r\). The positive correlation (positive value of \(r\)) results in a positive slope (upward trend), and a negative correlation (negative value of \(r\)) results in a negative slope (downward trend).

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Most popular questions from this chapter

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