Math Problem Statement

The Pearson correlation coefficient between two variables is: (Options include close to 1 if they tend to move in the same direction, close to 0 if they move independently, close to -1 if they move in opposite directions, and all of these.)

Solution

The correct answer is "All of these."

Explanation:

The Pearson correlation coefficient, denoted as rr, measures the linear relationship between two variables. Its value ranges from -1 to 1, with interpretations as follows:

  1. r1r \approx 1: Indicates a strong positive correlation; as one variable increases, the other also tends to increase.
  2. r1r \approx -1: Indicates a strong negative correlation; as one variable increases, the other tends to decrease.
  3. r0r \approx 0: Indicates no linear correlation; the variables move independently of each other.

Thus, all the options listed describe specific cases of the Pearson correlation coefficient.

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Math Problem Analysis

Mathematical Concepts

Statistics
Correlation
Linear Relationships

Formulas

Pearson correlation coefficient formula: r = Σ((xi - x̄)(yi - ȳ)) / (n * σx * σy)

Theorems

Properties of Pearson correlation

Suitable Grade Level

Grades 9-12