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Chapter 5: Problem 16

A sample of automobiles traversing a certain stretch of 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 traverse this segment of highway. Would the sample correlation coefficient be closest to \(.9, .3,-.3\), or \(-.9 ?\) Explain.

Short Answer

Expert verified
The sample correlation coefficient would be closest to -0.9. This is due to the strong negative correlation between the speed of an automobile and the time it takes to traverse a specific segment of highway.

Step by step solution

01

Understand the Concept of Correlation Coefficient

The correlation coefficient measures the degree of linear relationship between two variables. It ranges from -1 to 1. A correlation coefficient of -1 indicates a perfect negative correlation, where an increase in one variable corresponds to a decrease in the other. In contrast, a correlation coefficient of 1 indicates a perfect positive correlation, where both variables increase or decrease together. A coefficient close to 0 indicates no linear correlation.
02

Apply the Concept to the Current Problem

In this case, the speed of automobiles (\(x\)) and the time needed to traverse a certain distance (\(y\)) have an inverse relationship. This means when the speed increases, the time needed to traverse that distance decreases. Therefore, the correlation is expected to be negative.
03

Choose the Correct Correlation Coefficient from the Provided Choices

Given the options of .9, .3, -.3, and -.9, the correlation coefficient that best fits the relationship in this situation is -0.9. This is because -0.9 signifies a very strong negative relationship, which is what you would expect for speed and travel time: as speed increases, travel time decreases substantially.

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