Question

The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12. If \( \mu \) and \( \sigma^2 \) denote the mean and variance of the correct observations respectively, then \( 15(\mu + \mu^2 + \sigma^2) \) is equal to \(\ldots\)

Answer 1

Correct Answer: 2521

Let the incorrect mean be \(\mu'\) and standard deviation be \(\sigma'\).

We have:  
\(\mu' = \frac{\sum z_i}{15} = 12 \implies \sum z_i = 15 \times 12 = 180.\)

After correcting the value:  
\(\sum z_i = 180 - 10 + 12 = 182.\)

Corrected mean:
\(\mu = \frac{182}{15}.\)

Also:  
\(\sigma'^2 = \frac{\sum z_i^2}{15} - \mu'^2.\)

Given \(\sigma' = 3\):  
\(\sigma'^2 = 9 \implies \frac{\sum z_i^2}{15} - 9 = 9 \implies \sum z_i^2 = 15 \times 9 + 180^2.\)

Corrected variance:
\(\sigma^2 = 2339.\)

The required value is:  
\(15 \left(\mu^2 + \sigma^2 + \sigma^2\right) = 2521.\)

The Correct answer is: 2521