4. In order to find the correlation coefficient between the two variables X and Y from 12 pair of observations, the following calculations were made: \( \Sigma \mathrm{X}=30, \Sigma \mathrm{Y}=5, \Sigma \mathrm{X}^{2}=670, \Sigma \mathrm{Y}^{2}=285, \Sigma \mathrm{XY}=334 \) On subsequent verification, it was found that the pair \( (\mathrm{X}=11, \mathrm{Y}=4) \) was copied wrong, the correct value being \( (\mathrm{X}=10, \mathrm{Y}=14) \). Find the correct value of correlation coefficient.

We start with the given summary statistics from 12 pairs of observations: \[ \Sigma X = 30,\quad \Sigma Y = 5,\quad \Sigma X^2 = 670,\quad \Sigma Y^2 = 285,\quad \Sigma XY = 334. \] It was later discovered that the pair \((X=11,\,Y=4)\) was recorded in error, and the correct pair is \((X=10,\,Y=14)\). **Step 1. Correct the sums** For each statistic, we remove the contribution of the incorrect pair and add that of the correct pair. - **Sum of \(X\):** \[ \Sigma X_{\text{new}} = 30 - 11 + 10 = 29. \] - **Sum of \(Y\):** \[ \Sigma Y_{\text{new}} = 5 - 4 + 14 = 15. \] - **Sum of \(X^2\):** \[ \Sigma X^2_{\text{new}} = 670 - 11^2 + 10^2 = 670 - 121 + 100 = 649. \] - **Sum of \(Y^2\):** \[ \Sigma Y^2_{\text{new}} = 285 - 4^2 + 14^2 = 285 - 16 + 196 = 465. \] - **Sum of \(XY\):** \[ \Sigma XY_{\text{new}} = 334 - (11 \times 4) + (10 \times 14) = 334 - 44 + 140 = 430. \] **Step 2. Use the formula for the correlation coefficient** The correlation coefficient \( r \) is given by: \[ r = \frac{n\Sigma XY - (\Sigma X)(\Sigma Y)}{\sqrt{\left[n\Sigma X^2 - (\Sigma X)^2\right]\left[n\Sigma Y^2 - (\Sigma Y)^2\right]}}, \] where \( n = 12 \). Plug in the corrected sums: - Numerator: \[ n\Sigma XY - (\Sigma X)(\Sigma Y) = 12 \times 430 - 29 \times 15 = 5160 - 435 = 4725. \] - Denominator (first factor): \[ n\Sigma X^2 - (\Sigma X)^2 = 12 \times 649 - 29^2 = 7788 - 841 = 6947. \] - Denominator (second factor): \[ n\Sigma Y^2 - (\Sigma Y)^2 = 12 \times 465 - 15^2 = 5580 - 225 = 5355. \] Thus, the denominator becomes: \[ \sqrt{6947 \times 5355}. \] **Step 3. Compute the correlation coefficient** Expressing \( r \): \[ r = \frac{4725}{\sqrt{6947 \times 5355}}. \] First, compute the product inside the square root: \[ 6947 \times 5355 = 37\,201\,785. \] Then, take the square root: \[ \sqrt{37\,201\,785} \approx 6099. \] Thus, \[ r \approx \frac{4725}{6099} \approx 0.775. \] **Final Answer:** \[ r \approx 0.775. \]