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, \( r \), for the given father and son heights, we will use the formula:

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

where:
- \( X \) and \( Y \) are the individual values of the two variables (in this case, heights of fathers and sons),
- \( \bar{X} \) and \( \bar{Y} \) are the mean values of \( X \) and \( Y \), respectively.

Given dataTo calculate Pearson's correlation coefficient, \( r \), for the given father and son heights, we will use the formula:

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

where:
- \( X \) and \( Y \) are the individual values of the two variables (in this case, heights of fathers and sons),
- \( \bar{X} \) and \( \bar{Y} \) are the mean values of \( X \) and \( Y \), respectively.

Given 
- Heights of Fathers (X): 65, 66, 67, 68, 69, 70, 67, 72
- Heights of Sons (Y): 67, 68, 65, 68, 72, 72, 69, 71

### Step-by-Step Calculation:

1. **Calculate the Means** \( \bar{X} \) and \( \bar{Y} \).
   
2. **Find the Deviations** from the means for each \( X \) and \( Y \) (i.e., \( X - \bar{X} \) and \( Y - \bar{Y} \)).

3. **Calculate the Products** of these deviations \( (X - \bar{X})(Y - \bar{Y}) \).

4. **Sum the Squares** of the deviations for both \( X \) and \( Y \) values.

5. **Apply the Formula** to find \( r \).

I’ll calculate these steps for you.The Pearson's correlation coefficient \( r \) for the given heights of fathers and sons is approximately \( 0.729 \). This indicates a moderately strong positive correlation, suggesting that as the height of the father increases, the height of the son tends to increase as well.

Would you like a detailed breakdown of each calculation step?

Here are five additional questions you might consider:

1. How does Pearson's correlation coefficient interpret negative values?
2. What is the difference between correlation and causation?
3. What other types of correlation coefficients exist for non-linear data?
4. How sensitive is Pearson's correlation to outliers?
5. What sample size is ideal for calculating reliable correlation coefficients?

**Tip:** Pearson's correlation is best suited for linear relationships and may not accurately reflect non-linear patterns 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 how to calculate Pearson's correlation coefficient for the heights of fathers and sons, a statistical method that determines the relationship between two variables. Using given data sets for father and son heights, understand step-by-step how to apply Pearson's formula and interpret the results, suitable for Grades 10-12.

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