\( \mathrm{Y}: \mid \) Arithmetic means of X and Y series are 6 and 8 respectively. In order to find the correlation coefficient between the two variables X and Y from 12 pair of observations, the following calculations were made: \( \Sigma \mathrm{X}=30, \quad \Sigma \mathrm{Y}=5, \quad \Sigma \mathrm{X}^{2}=670, \quad \Sigma \mathrm{Y}^{2}=285, \quad \Sigma \mathrm{XY}=334 \) 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.

To find the correct correlation coefficient, we first need to adjust the given summation values by substituting the incorrect pair \((X=11, Y=4)\) with the correct pair \((X=10, Y=14)\). ### Step 1: Adjust the summations - Original summations: - \( \Sigma X = 30 \) - \( \Sigma Y = 5 \) - \( \Sigma X^2 = 670 \) - \( \Sigma Y^2 = 285 \) - \( \Sigma XY = 334 \) - Change from incorrect to correct pair: - For \(X\): Replace \(11\) with \(10\): \[ \Sigma X_{new} = 30 - 11 + 10 = 29 \] - For \(Y\): Replace \(4\) with \(14\): \[ \Sigma Y_{new} = 5 - 4 + 14 = 15 \] - For \(X^2\): Replace \(11^2\) with \(10^2\): \[ \Sigma X^2_{new} = 670 - 121 + 100 = 649 \] - For \(Y^2\): Replace \(4^2\) with \(14^2\): \[ \Sigma Y^2_{new} = 285 - 16 + 196 = 465 \] - For \(XY\): Replace \(11 \cdot 4\) with \(10 \cdot 14\): \[ \Sigma XY_{new} = 334 - 44 + 140 = 430 \] ### Step 2: Calculate the correlation coefficient The formula for the correlation coefficient \( r \) is given by: \[ r = \frac{n (\Sigma XY) - (\Sigma X)(\Sigma Y)}{\sqrt{[n \Sigma X^2 - (\Sigma X)^2][n \Sigma Y^2 - (\Sigma Y)^2]}} \] where \( n = 12 \). Substituting in our adjusted values: \[ r = \frac{12 \cdot 430 - 29 \cdot 15}{\sqrt{[12 \cdot 649 - 29^2][12 \cdot 465 - 15^2]}} \] Calculating the numerator: \[ = 12 \cdot 430 - 29 \cdot 15 = 5160 - 435 = 4725 \] Calculating the first part of the denominator: \[ 12 \cdot 649 - 29^2 = 7788 - 841 = 6947 \] Calculating the second part of the denominator: \[ 12 \cdot 465 - 15^2 = 5580 - 225 = 5355 \] Thus the denominator becomes: \[ \sqrt{6947 \cdot 5355} \] Calculating this value: \[ \sqrt{6947 \cdot 5355} \approx \sqrt{37248785} \approx 6104.53 \] ### Final calculation of \( r \): \[ r \approx \frac{4725}{6104.53} \approx 0.773 \] Thus, the correct value of the correlation coefficient is approximately \( r \approx 0.773\).