Let's Solve the Problem Step-by-Step:

Part (i): Calculate the coefficient of correlation $r$

Given Data:

• $n = 10$
• $\Sigma X = 100$
• $\Sigma Y = 150$
• $\Sigma (X - 10)^2 = 180$
• $\Sigma (Y - 15)^2 = 215$
• $\Sigma (X - 10)(Y - 15) = 60$

Formula for Karl Pearson’s Coefficient of Correlation:

$r = \frac{\Sigma (X - \bar{X})(Y - \bar{Y})}{\sqrt{\Sigma (X - \bar{X})^2 \cdot \Sigma (Y - \bar{Y})^2}}$

Substitute Values:

• $\Sigma (X - \bar{X})(Y - \bar{Y}) = 60$
• $\Sigma (X - \bar{X})^2 = 180$
• $\Sigma (Y - \bar{Y})^2 = 215$

$r = \frac{60}{\sqrt{180 \cdot 215}}$

Calculate:

• $\sqrt{180 \cdot 215} = \sqrt{38700} = 196.68$
• $r = \frac{60}{196.68} = 0.305$

Part (ii): Calculate Karl Pearson’s Coefficient of Correlation

Given Data:

• $n = 12$
• $\Sigma X = 120$
• $\Sigma Y = 130$
• $\Sigma (X - 8)^2 = 150$
• $\Sigma (Y - 10)^2 = 200$
• $\Sigma (X - 8)(Y - 10) = 50$

Formula for $r$:

$r = \frac{\Sigma (X - 8)(Y - 10)}{\sqrt{\Sigma (X - 8)^2 \cdot \Sigma (Y - 10)^2}}$

Substitute Values:

• $\Sigma (X - 8)(Y - 10) = 50$
• $\Sigma (X - 8)^2 = 150$
• $\Sigma (Y - 10)^2 = 200$

$r = \frac{50}{\sqrt{150 \cdot 200}}$

Calculate:

• $\sqrt{150 \cdot 200} = \sqrt{30000} = 173.21$
• $r = \frac{50}{173.21} = 0.289$

Final Answers:

1. Part (i): $r = 0.305$
2. Part (ii): $r = 0.289$

Let me know if you'd like further clarifications or detailed steps for any part!

Questions to Expand Understanding:

1. What is the significance of Karl Pearson’s coefficient of correlation in statistics?
2. How does the sign of $r$ affect the interpretation of correlation between variables?
3. What are the key assumptions for using the Pearson correlation coefficient?
4. Can correlation imply causation? Why or why not?
5. How would you compute $r$ if you were only given raw data instead of summaries?

Tip:

Always check for outliers in your data, as they can significantly influence the Pearson correlation coefficient and lead to misleading results.