Math Problem Statement

After analyzing a set of paired quantitative data values, you determine that r=0.998 . What can be said about the linear relationship between the paired data values?

Solution

The value of the correlation coefficient r=0.998r = 0.998 indicates a very strong positive linear relationship between the paired data values. Here's what this implies:

  1. Direction of the relationship: Since rr is positive, the relationship is direct. As one variable increases, the other also tends to increase.

  2. Strength of the relationship: The correlation coefficient rr can range from -1 to 1. A value of 0.9980.998 is extremely close to 1, indicating that the relationship is very strong and nearly perfect.

  3. Linearity: Given how close rr is to 1, the data points lie very close to a straight line. In practical terms, a linear model would describe the relationship between the two variables with high accuracy.

  4. Predictability: With such a high rr value, predictions based on the linear relationship between the variables would be highly reliable.

  5. Causation: It's important to remember that correlation does not imply causation. The strong correlation indicates a strong association but doesn't necessarily mean that one variable causes changes in the other.

Would you like more details on any of these points?


Here are 5 related questions to expand on this:

  1. What is the significance of the coefficient of determination r2r^2 in this context?
  2. How would a correlation of r=0.998r = -0.998 differ in interpretation?
  3. How do outliers affect the value of rr?
  4. What are the assumptions behind using the Pearson correlation coefficient rr?
  5. How do you interpret r=0r = 0, and does that always mean no relationship?

Tip: Always visualize your data (e.g., using a scatter plot) before drawing conclusions from the correlation coefficient alone.

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Math Problem Analysis

Mathematical Concepts

Statistics
Correlation
Linear Relationships

Formulas

Pearson correlation coefficient r

Theorems

Properties of Correlation Coefficients

Suitable Grade Level

Grades 10-12