Question

Let the mean and variance of 12 observations be \( \frac{9}{2} \) and 4, respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is \( \frac{m}{n} \), where \( m \) and \( n \) are coprime, then \( m + n \) is equal to:

• A. 314
• B. 315
• C. 316
• D. 317

Hint:

For corrections in statistics, update both the sum and the sum of squares carefully, and recompute the variance using the corrected values.

Answer 1

The Correct Option is D

The mean of the 12 observations is given as:

\[ \frac{\Sigma x}{12} = \frac{9}{2} \implies \Sigma x = 54. \]

The variance of the 12 observations is given as:

\[ \frac{\Sigma x^2}{12} - \left(\frac{\Sigma x}{12}\right)^2 = 4. \]

Substitute \( \Sigma x = 54 \):

\[ \frac{\Sigma x^2}{12} - \left(\frac{9}{2}\right)^2 = 4 \implies \frac{\Sigma x^2}{12} - \frac{81}{4} = 4. \]

Simplify:

\[ \Sigma x^2 = 12 \left(4 + \frac{81}{4}\right) = 12 \cdot \frac{97}{4} = 291. \]

After correction, the observations 9 and 10 are replaced with 7 and 14.

The corrected sum of the observations is:

\[ \Sigma x_{\text{new}} = 54 - (9 + 10) + (7 + 14) = 56. \]

The corrected sum of squares is:

\[ \Sigma x^2_{\text{new}} = 291 - (81 + 100) + (49 + 196) = 355. \]

The corrected variance is:

\[ \sigma^2_{\text{new}} = \frac{\Sigma x^2_{\text{new}}}{12} - \left(\frac{\Sigma x_{\text{new}}}{12}\right)^2. \]

Substitute \( \Sigma x^2_{\text{new}} = 355 \) and \( \Sigma x_{\text{new}} = 56 \):

\[ \sigma^2_{\text{new}} = \frac{355}{12} - \left(\frac{56}{12}\right)^2. \]

Simplify each term:

• \(\frac{355}{12}\) remains as is.
• \(\left(\frac{56}{12}\right)^2 = \frac{(56)^2}{144} = \frac{3136}{144} = \frac{49}{36}.\)

Thus:

\[ \sigma^2_{\text{new}} = \frac{355}{12} - \frac{49}{36}. \]

Take the LCM of 12 and 36:

\[ \sigma^2_{\text{new}} = \frac{1065}{36} - \frac{49}{36} = \frac{1016}{36}. \]

Simplify the fraction:

\[ \sigma^2_{\text{new}} = \frac{254}{9}. \]

Here, \( m = 254 \) and \( n = 9 \), which are coprime.

Finally, \( m + n = 254 + 9 = 317 \).