Math Problem Statement

The weight (in pounds) and height (in inches) for a child were measured every few months over a two-year period. Using technology, what is the correlation coefficient?

Solution

To find the correlation coefficient for the data in the table, we can use a statistical calculator or software to calculate the correlation coefficient (denoted as rr) for the pairs of weight (xx) and height (yy) values.

The data pairs are:

  • xx (Weight): 8, 12, 18, 24, 30, 32, 35, 37, 40
  • yy (Height): 22, 23, 26, 30, 32, 33, 35, 36, 38

Steps to Calculate the Correlation Coefficient rr:

  1. Compute the mean of xx values and the mean of yy values.
  2. Calculate the deviation of each xx and yy value from their respective means.
  3. Find the product of these deviations for each data pair.
  4. Calculate the square of each deviation for both xx and yy.
  5. Use the correlation formula:

r=(xxˉ)(yyˉ)(xxˉ)2(yyˉ)2r = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \cdot \sum (y - \bar{y})^2}}

After using technology or calculator to compute this, we get a correlation coefficient r0.997r \approx 0.997.

Thus, the correct answer is:

  • 0.997

Would you like a detailed breakdown of these calculations?

Further Questions

  1. What does a correlation coefficient close to 1 imply about the relationship between weight and height in this data?
  2. How would you interpret a correlation coefficient of -0.503 in a similar context?
  3. Can you identify any outliers in this data set based on the correlation coefficient?
  4. How would the correlation coefficient change if there were an error in one of the data points?
  5. Why is it important to verify the assumptions of linearity before relying on the correlation coefficient?

Tip

Always interpret correlation in the context of the data. A high correlation does not imply causation but indicates a strong association between variables.

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Math Problem Analysis

Mathematical Concepts

Statistics
Correlation
Linear Relationships

Formulas

Correlation coefficient formula: r = (Σ(x - x̄)(y - ȳ)) / √(Σ(x - x̄)^2 * Σ(y - ȳ)^2)

Theorems

Pearson Correlation Coefficient

Suitable Grade Level

Grades 10-12