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 indicates a very strong positive linear relationship between the paired data values. Here's what this implies:
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Direction of the relationship: Since is positive, the relationship is direct. As one variable increases, the other also tends to increase.
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Strength of the relationship: The correlation coefficient can range from -1 to 1. A value of is extremely close to 1, indicating that the relationship is very strong and nearly perfect.
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Linearity: Given how close 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.
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Predictability: With such a high value, predictions based on the linear relationship between the variables would be highly reliable.
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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:
- What is the significance of the coefficient of determination in this context?
- How would a correlation of differ in interpretation?
- How do outliers affect the value of ?
- What are the assumptions behind using the Pearson correlation coefficient ?
- How do you interpret , 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
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