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Chapter 5: Problem 2

Is the following statement correct? Explain why or why not. A correlation coefficient of 0 implies that no relationship exists between the two variables under study.

Short Answer

Expert verified
No, the statement is not completely correct. A correlation coefficient of 0 implies no linear relationship, not necessarily no relationship at all. Other relationships, possibly nonlinear, might still exist between the two variables.

Step by step solution

01

Clarifying Correlation Coefficient

The correlation coefficient measures the extent to which two variables tend to change together. The coefficient describes both the strength and the direction of the relationship. The correlation coefficient ranges from -1 to 1. If the correlation coefficient is close to -1, it suggests a strong negative relationship. A coefficient close to 1 suggests a strong positive relationship. A coefficient close to 0 suggests no linear relationship.
02

Understanding Zero Correlation

When the correlation coefficient is 0, it means there is no linear relationship between the variables. This does not necessarily mean that the variables are not related at all, only that they do not move in tandem in a linear way. For instance, there may be a complex nonlinear relationship that the correlation coefficient won't detect.
03

Analyze the Statement

Given the above information, the statement 'A correlation coefficient of 0 implies that no relationship exists between the two variables under study.' can be incorrect. A correlation coefficient of 0 implies no linear relationship, not necessarily no relationship at all.

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Most popular questions from this chapter

The article "The Epiphytic Lichen Hypogymnia physodes as a Bioindicator of Atmospheric Nitrogen and Sulphur Deposition in Norway" (Environmental Monitoring and Assessment [1993]: \(27-47\) ) gives the following data (read from a graph in the paper) on \(x=\mathrm{NO}_{3}\) wet deposition (in grams per cubic meter) and \(y=\) lichen (\% dry weight): a. What is the equation of the least-squares regression line? \(\quad \hat{y}=0.3651+0.9668 \mathrm{x}\) b. Predict lichen dry weight percentage for an \(\mathrm{NO}_{3}\) depo sition of \(0.5 \mathrm{~g} / \mathrm{m}^{3}\).

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For each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to \(0 .\) Explain your choice. a. Maximum daily temperature and cooling costs b. Interest rate and number of loan applications c. Incomes of husbands and wives when both have fulltime jobs d. Height and IQ e. Height and shoe size f. Score on the math section of the SAT exam and score on the verbal section of the same test g. Time spent on homework and time spent watching television during the same day by elementary school children h. Amount of fertilizer used per acre and crop yield (Hint: As the amount of fertilizer is increased, yield tends to increase for a while but then tends to start decreasing.)

The accompanying data represent \(x=\) the amount of catalyst added to accelerate a chemical reaction and \(y=\) the resulting reaction time: $$ \begin{array}{rrrrrr} x & 1 & 2 & 3 & 4 & 5 \\ y & 49 & 46 & 41 & 34 & 25 \end{array} $$ a. Calculate \(r\). Does the value of \(r\) suggest a strong linear relationship? b. Construct a scatterplot. From the plot, does the word linear really provide the most effective description of the relationship between \(x\) and \(y\) ? Explain.

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.
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