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 (R2R^2), 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:

  • nn is the number of data points (5 in this case)
  • xx represents the age values
  • yy represents the bone density values

2. Find the coefficient of determination:

Once we have rr, the coefficient of determination is simply:

R2=r2R^2 = r^2

Let’s compute the required values step-by-step:

  • xx = [39, 40, 64, 66, 70] (Age values)
  • yy = [353, 329, 322, 316, 313] (Bone density values)

We will first compute the necessary summations and then proceed to find rr and finally R2R^2.

3. Summations:

We will calculate the following values:

  • x\sum x
  • y\sum y
  • x2\sum x^2
  • y2\sum y^2
  • xy\sum xy

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 (R2R^2) 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:

  1. How does the value of R2R^2 indicate the strength of the relationship between age and bone density?
  2. What does a low R2R^2 value suggest about the accuracy of predictions made using the regression line?
  3. How would you interpret the significance of the correlation coefficient (rr) in this case?
  4. Can R2R^2 ever be greater than 1? If so, under what circumstances?
  5. How do outliers affect the calculation of the correlation coefficient and coefficient of determination?

Tip: To assess the statistical significance of rr, you would typically perform a hypothesis test (e.g., using a t-test for correlation) to check whether rr 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