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Chapter 13: Problem 62

Can the values of \(B\) and \(\rho\) calculated for the same population data have different signs? Explain.

Short Answer

Expert verified
Yes, the values of \(B\) and \(\rho\) calculated for the same population data can have different signs. This will typically happen when there are multiple independent variables in the data and some have inverse impacts on the dependent variable.

Step by step solution

01

Understanding Variables

First, understand the nature of the both variables. The coefficient \(B\) in a linear regression model indicates the effect that a change in one unit of the independent variable will have on the dependent variable. It allows for linear change which can be either positive or negative depicted by the sign accompanying the coefficient.
02

Understanding Correlation

The correlation coefficient \(\rho\), on the other hand, measures the linear correlation between two variables. It ranges between -1 and +1, with -1 indicating a perfect negative linear correlation, +1 a perfect positive linear correlation and 0 no linear correlation.
03

Compare and Conclude

Comparing the two, one can see that while both relate to linear relationships, \(B\) refers to how a unit increase in one variable affects the other, while \(\rho\) indicates the directional relationship. Thus, it's possible for \(B\) and \(\rho\) to have different signs in certain scenarios, especially in the presence of multiple independent variables where each could be impacting the dependent variable inversely.

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Most popular questions from this chapter

Construct a \(95 \%\) confidence interval for the mean value of \(y\) and a \(95 \%\) prediction interval for the predicted value of \(y\) for the following. a. \(\hat{y}=13.40+2.58 x\) for \(x=8\) given \(s_{e}=1.29, \bar{x}=11.30, \mathrm{SS}_{x x}=210.45\), and \(n=12\) b. \(\hat{y}=-8.6+3.72 x\) for \(x=24\) given \(s_{e}=1.89, \bar{x}=19.70, \mathrm{SS}_{x x}=315.40\), and \(n=10\)

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Consider the formulas for calculating a prediction interval for a new (specific) value of \(y\). For each of the changes mentioned in parts a through \(\mathrm{c}\) below, state the effect on the width of the confidence interval (increase, decrease, or no change) and why it happens. Note that besides the change mentioned in each part, everything else such as the values of \(a, b, \bar{x}, s_{e}\), and \(S S_{x x}\) remains unchanged. a. The confidence level is increased. b. The sample size is increased. c. The value of \(x_{0}\) is moved farther away from the value of \(\bar{x}\). d. What will the value of the margin of error be if \(x_{0}\) equals \(\bar{x}\) ?

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