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 39 40 64 66 70 Bone Density 353 329 322 316 313 Table
Step 6 of 6 : Find the value of the coefficient of determination. Round your answer to three decimal places.
Solution
To calculate the coefficient of determination (), we need to follow these steps:
1. Calculate the correlation coefficient (r):
The formula for the correlation coefficient is:
[ r = \frac{n \sum xy - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} ]
Where:
- is the number of data points (5 in this case)
- represents the age values
- represents the bone density values
2. Find the coefficient of determination:
Once we have , the coefficient of determination is simply:
Let’s compute the required values step-by-step:
- = [39, 40, 64, 66, 70] (Age values)
- = [353, 329, 322, 316, 313] (Bone density values)
We will first compute the necessary summations and then proceed to find and finally .
3. Summations:
We will calculate the following values:
Let's proceed with these calculations. numpy as np
Given data
x = np.array([39, 40, 64, 66, 70]) y = np.array([353, 329, 322, 316, 313])
Number of data points
n = len(x)
Summations
sum_x = np.sum(x) sum_y = np.sum(y) sum_x2 = np.sum(x2) sum_y2 = np.sum(y2) sum_xy = np.sum(x * y)
Calculate the correlation coefficient (r)
r = (n * sum_xy - sum_x * sum_y) / np.sqrt((n * sum_x2 - sum_x2) * (n * sum_y2 - sum_y2))
Coefficient of determination (R^2)
R_squared = r**2 R_squared_rounded = round(R_squared, 3)
R_squared_roundedThe value of the coefficient of determination () for the given data is 0.728, rounded to three decimal places.
Would you like further details on this process or have any questions about the calculation?
Here are 5 related questions you might find interesting:
- How does the value of indicate the strength of the relationship between age and bone density?
- What does a low value suggest about the accuracy of predictions made using the regression line?
- How would you interpret the significance of the correlation coefficient () in this case?
- Can ever be greater than 1? If so, under what circumstances?
- How do outliers affect the calculation of the correlation coefficient and coefficient of determination?
Tip: To assess the statistical significance of , you would typically perform a hypothesis test (e.g., using a t-test for correlation) to check whether is significantly different from zero.
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Math Problem Analysis
Mathematical Concepts
Regression Analysis
Correlation Coefficient
Coefficient of Determination
Statistics
Formulas
Regression Line: yˆ = b0 + b1x
Correlation Coefficient (r): r = (n * Σxy - Σx * Σy) / √[(n * Σx² - (Σx)²)(n * Σy² - (Σy)²)]
Coefficient of Determination (R²): R² = r²
Theorems
Properties of the Correlation Coefficient
Interpretation of Coefficient of Determination
Suitable Grade Level
Grades 11-12
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