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QuestionWhat value of rr indicates one variable "explains" less than 50%50\% of the other? Options: A. r=−1r=-1, B. r=−0.9r=-0.9, C. r=−0.7r=-0.7, D. r=0.9r=0.9.

Studdy Solution

STEP 1

Assumptions1. The correlation coefficient rr is a measure of the linear relationship between two variables. . The value of rr^{}, also known as the coefficient of determination, measures how much of the variance in one variable is explained by the variance in the other variable.
3. r=0r^{}=0 means that one variable does not explain any of the variance in the other variable.
4. r=1r^{}=1 means that one variable explains all of the variance in the other variable.
5. We are looking for a value of rr such that rr^{} is less than 0.50.5, meaning that less than50% of the variance in one variable is explained by the other variable.

STEP 2

We need to find a value of rr such that r2r^{2} is less than 0.50.5. We can do this by setting up the inequality r2<0.5r^{2}<0.5 and solving for rr.
r2<0.5r^{2}<0.5

STEP 3

To solve for rr, we take the square root of both sides of the inequality. Note that when we take the square root of both sides of an inequality, we must consider both the positive and negative roots.
r<0.5andr>−0.5r<\sqrt{0.5} \quad \text{and} \quad r>-\sqrt{0.5}

STEP 4

Calculate the square root of0..
r<0.≈0.707andr>−0.≈−0.707r<\sqrt{0.} \approx0.707 \quad \text{and} \quad r>-\sqrt{0.} \approx -0.707

STEP 5

From the given options, we see that the values of rr that satisfy the inequality r<0.707r<0.707 and r>−0.707r>-0.707 are r=−0.9r=-0.9 and r=−0.7r=-0.7.
Therefore, the values of rr that correspond to a situation in which one variable "explains" less than 50%50 \% of the other variable are r=−0.9r=-0.9 and r=−0.7r=-0.7.

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