Cskuste de Kal Pearson's Correlation Coefficient for the Following Heights (in inches) of Fathers(X) and Sons (Y)

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To calculate Pearson's correlation coefficient \( r \) between the heights of fathers (X) and sons (Y), we use the following formula:

\[
r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum (X_i - \bar{X})^2 \sum (Y_i - \bar{Y})^2}}
\]

where:
- \( X_i \) and \( Y_i \) are the individual values of the fathers' and sons' heights.
- \( \bar{X} \) and \( \bar{Y} \) are the mean values of X and Y, respectively.

Let's go through each step to find \( r \).

### Step 1: Compute the means of X and Y
1. **Calculate the mean of \( X \) (fathers' heights):**
   \[
   \bar{X} = \frac{65 + 66 + 67 + 68 + 69 + 70 + 67 + 72}{8}
   \]

2. **Calculate the mean of \( Y \) (sons' heights):**
   \[
   \bar{Y} = \frac{67 + 68 + 65 + 68 + 72 + 72 + 69 + 71}{8}
   \]

### Step 2: Calculate the deviations from the mean for each \( X \) and \( Y \)
For each value in X and Y, compute:
   - \( X_i - \bar{X} \)
   - \( Y_i - \bar{Y} \)

### Step 3: Compute \( (X_i - \bar{X})(Y_i - \bar{Y}) \), \( (X_i - \bar{X})^2 \), and \( (Y_i - \bar{Y})^2 \) for each pair

### Step 4: Substitute these values into the Pearson's formula and calculate \( r \)

Let’s calculate these values to find the correlation coefficient.The Pearson's correlation coefficient \( r \) for the given heights of fathers and sons is approximately \( 0.73 \). This indicates a moderate to strong positive correlation, suggesting that as the height of fathers increases, the height of their sons tends to increase as well.

Would you like further details on any of the calculation steps?

### Additional Questions
1. How does Pearson’s correlation coefficient relate to causation between variables?
2. What are other ways to measure correlation, besides Pearson’s coefficient?
3. How does the size of the data sample affect the reliability of Pearson’s correlation?
4. In what scenarios would a Pearson correlation of 0.73 indicate a weak relationship?
5. What steps would change if calculating correlation on larger datasets?

**Tip:** Always check the data distribution before applying Pearson’s correlation; it assumes a linear relationship between variables.

Cskuste de Kal Pearson's Correlation Coefficient for the Following Heights (in inches) of Fathers (X) and Sons (Y): X = [65, 66, 67, 68, 69, 70, 67, 72], Y = [67, 68, 65, 68, 72, 72, 69, 71]

Learn to calculate Pearson's correlation coefficient with a step-by-step example using father-son height data. This problem covers key concepts in statistics, including calculating means, deviations, and correlation. Suitable for students in Grades 10-12.

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