The mean and variance of six observations are 8 and 16 respectively. If each observation is multiplied by 3, find the new mean and new variance of the resulting observations.

The mean and variance of six observations are 8 and 16 respectively. I

1.	The mean and variance of six observations are 8 and 16 respectively. If each observation is multiplied by 3, find the new mean and new variance of the resulting observations.
Answer» SOLUTION :Let the given observations `x_(1), x_(2), x_(3), x_(4), x_(5), x_(6).` Then, mean `=8 rArr (1)/(6)(x_(1)+x_(2)+x_(3)+x_(4)+x_(5)+x_(6))=8` `rArr" "x_(1)+x_(2)+x_(3)+x_(4)+x_(5)+x_(6)=48."(i)"` Also, variance =16 `rArr" "(1)/(6)(x_(1)^(2)+x_(2)^(2)+x_(3)^(2)+x_(4)^(2)+x_(5)^(2)+x_(6)^(2))-8^(2)=16" "[because sigma^(2)=(Sigmax_(i)^(2))/(n)-(barx)^(2)]` `rArr" "x_(1)^(2)+x_(2)^(2)+x_(3)^(2)+x_(4)^(2)+x_(5)^(2)+x_(6)^(2)=480."...(ii)"` When each observation is multiplied by 3, then new observations are `3x_(1), 3x_(2), 3x_(3), 3x_(4), 3x_(5) and 3x_(6).` `THEREFORE" new mean "=(1)/(6)(3x_(1)+3x_(2)+3x_(4)+3x_(5)+3x_(6))` `=(3)/(6)(x_(1)+x_(2)+x_(3)+x_(4)+x_(5)+x_(6))=((1)/(2)xx48)=24" [using (i)]"` `therefore" new variance "((3x_(1))^(2)+(3x_(2))^(2)+(3x_(3))^(2)+(3x_(4))^(2)+(3x_(5))^(2)+(3x_(6))^(2))/(6)-(24)^(2)` `=(9)/(6)(x_(1)^(2)+x_(2)^(2)+x_(3)^(2)+x_(4)^(2)+x_(5)^(2)+x_(6)^(2))-576` `=((9)/(6)xx480)-576=(720-576)=144" [using (ii)]".` Hence, new mean = 24 and new variance `= 144.`