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The variance of 10 observation is 16 and their mean is 12. If each observation is multiplied by 4, what are the new mean and the new variance?

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Best answer

Given, 

N = 10,

σ = 16, and 
\(\bar{x}\) = 12.

Let the new mean be \(\bar{x_1}\) and new variance be σ12.

Here, 

4\(\bar{x}\) = 4\(\bar{x}\)

We Know,

\(σ_1^2=\frac{\sum {x'}_i^2}{N}-(\frac{\sum \bar{x}'}{N})^2\)

\(=\frac{\sum (4x_i)^2}{N}-(\frac{\sum \bar{4x_i}}{N})^2\)

\(=16\{\frac{\sum x_i^2}{N}-(\frac{\sum {x_i}}{N})^2\}\)

= 16 × σ2

= 16 × 16

= 256

Also,

\(\bar{x_1}=\frac{\sum x_i}{N}\)

\(=4\frac{\sum x_i}{N}\)

\(=4\bar{x}\)

\(=\frac{\sum 4x_i}{N}\)

= 4 x 12

= 48

∴ New mean = 48 and new variance = 256.

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