Math Problem Statement

The table below gives the age and bone density for five randomly selected women. Using this data, consider the equation of the regression line, yˆ=b0+b1x , for predicting a woman's bone density based on her age. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant.

Age 47 49 51 58 63 Bone Density 360 353 336 333 332

Find the value of the coefficient of determination. Round your answer to three decimal places.

Solution

The coefficient of determination (R-squared) is approximately 0.700, rounded to three decimal places. This indicates that about 70.0% of the variability in bone density is explained by the age of the women in this dataset.

Would you like more details or have any other questions?

Here are some related questions:

  1. How do we interpret the R-squared value in context?
  2. What does it mean if the correlation coefficient is not statistically significant?
  3. How can we test if the correlation coefficient is statistically significant?
  4. How is the regression line constructed from the data?
  5. What factors could affect the relationship between age and bone density?

Tip: R-squared values closer to 1 indicate a better fit, but even a high R-squared doesn't guarantee causality.

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Math Problem Analysis

Mathematical Concepts

Statistics
Regression Analysis
Correlation
Coefficient of Determination (R-squared)

Formulas

yˆ = b0 + b1x (Equation of a regression line)
R^2 = 1 - (SS_res / SS_tot) (Coefficient of Determination)

Theorems

Linear Regression Theorem
Pearson Correlation Coefficient

Suitable Grade Level

Grades 10-12

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