Math Problem Statement
Solution
Let's break down the problem step by step.
-
Interpret the r-value:
- The -value provided is . Since the -value (correlation coefficient) ranges between -1 and 1, a value of indicates a strong positive linear correlation because it is close to 1.
- So, the correct answer for the first part of the question is:
"There is a strong positive linear correlation."
-
Predict the age of a tree with a diameter of 6 inches:
- The regression equation is given as:
where:
- is the diameter of the tree (in inches), and
- is the predicted age of the tree (in years).
To predict the age of a tree with a diameter of 6 inches, substitute into the equation: Therefore, the predicted age is approximately 41.34 years.
- The regression equation is given as:
where:
-
Classify the prediction:
- Since the prediction is being made for a diameter of 6 inches, and the dataset includes trees with diameters between 3 and 16 inches, this prediction falls within the given range. Thus, this is an interpolation (since interpolation refers to predicting values within the range of the data).
Summary of answers:
- Strong positive linear correlation.
- Predicted age for a 6-inch diameter tree: 41.34 years.
- This is interpolation.
Do you want more details or have any questions? Here are five related questions for deeper understanding:
- How does the -value (0.743) relate to the quality of the regression model?
- What would the prediction be for a tree with a diameter outside the given range (e.g., 18 inches)?
- Can the regression model accurately predict extreme values like for very small or large trees?
- What does the intercept (1.022) mean in the context of this problem?
- How would the model change if the relationship was non-linear?
Tip: The closer the correlation coefficient is to 1 or -1, the stronger the linear relationship between the variables!
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Correlation
Interpolation
Extrapolation
Formulas
y = ax + b
r (correlation coefficient)
Theorems
Linear Regression Theorem
Correlation Theorem
Suitable Grade Level
Grades 9-12
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