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.

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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:**
$$
r \approx 0.775.
$$
The correct correlation coefficient is approximately 0.775.

In order to find the correlation coefficient between the two variables X and Y from 12 pair
of observations, the following calculations were made:

On subsequent verification, it was found that the pair was copied wrong,
the correct value being . Find the correct value of correlation coefficient.

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In order to find the correlation coefficient between the two variables X and Y from 12 pair of observations, the following calculations were made: On subsequent verification, it was found that the pair was copied wrong, the correct value being . Find the correct value of correlation coefficient.