Correct Answer - Option 1 : 400
Given, mean and standard deviation of the 50 observations are equal to 16.
Mean, \(\left( \mu \right) = \frac{{\sum {x_i}}}{{50}} = \frac{{{x_1} + {x_2} + \ldots .{x_{50}}}}{{50}} = 16\)
∴ ∑xi = 16 × 50
Standard deviation, \(\left( \sigma \right) = \sqrt {\frac{{\sum x_i^2}}{{50}} - {{(\mu )}^2}} = 16\)
\( \Rightarrow \frac{{\sum x_i^2}}{{50}} = \)256 × 2
Required mean \( = \frac{{{{\left( {{x_1} - 4} \right)}^2} + {{\left( {{x_2} - 4} \right)}^2} + \ldots {{\left( {{x_{50}} - 4} \right)}^2}}}{{50}}\)
\( = \frac{{\sum {{\left( {{x_i} - 4} \right)}^2}}}{{50}}\)
\( = \frac{{\sum x_i^2 + 16 \times 50 - 8\sum {x_i}}}{{50}}\)
= 256 × 2 + 16 - 8 × 16
= 528 - 128 = 400