Math Problem Statement

Data was collected from a sample of 54 trees with diameters between 3 and 16 inches. A regression analysis was run to determine if there is a relationship between the diameter of a tree (x, in inches) and the tree's age (y, in years). The results of the regression are given below. Predict the age of a tree with a 6-inch diameter and classify this prediction as interpolation or extrapolation.

Solution

Let's break down the problem step by step.

  1. Interpret the r-value:

    • The rr-value provided is r=0.862r = 0.862. Since the rr-value (correlation coefficient) ranges between -1 and 1, a value of r=0.862r = 0.862 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."
  2. Predict the age of a tree with a diameter of 6 inches:

    • The regression equation is given as: y=ax+by = ax + b where:
      • a=6.72a = 6.72
      • b=1.022b = 1.022
      • xx is the diameter of the tree (in inches), and
      • yy is the predicted age of the tree (in years).

    To predict the age of a tree with a diameter of 6 inches, substitute x=6x = 6 into the equation: y=6.72(6)+1.022y = 6.72(6) + 1.022 y=40.32+1.022=41.342y = 40.32 + 1.022 = 41.342 Therefore, the predicted age is approximately 41.34 years.

  3. 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:

  1. Strong positive linear correlation.
  2. Predicted age for a 6-inch diameter tree: 41.34 years.
  3. This is interpolation.

Do you want more details or have any questions? Here are five related questions for deeper understanding:

  1. How does the r2r^2-value (0.743) relate to the quality of the regression model?
  2. What would the prediction be for a tree with a diameter outside the given range (e.g., 18 inches)?
  3. Can the regression model accurately predict extreme values like for very small or large trees?
  4. What does the intercept bb (1.022) mean in the context of this problem?
  5. How would the model change if the relationship was non-linear?

Tip: The closer the correlation coefficient rr 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