The mean of 5 observations is 15 and variance is 9. If two observations having values -5 and 13 are combined with these observations, then what will be the new variance?

Step-by-step solution

Let the observations are (x₁, x₂, x₃, x₄, x₅ ), so the mean ( x = x₁+x₂+x₃+x₄+x₅ 5 =15 ) (given) ( x₁+x₂+x₃+x₄+x₅=75 ) ...(i) and variance ( aligned & ^2= x₁^2+x₂^2+x₃^2+x₄^2+x₅^2 5 -( x )^2=9 (given) & x₁^2+x₂^2+x₃^2+x₄^2+x₅^2=5[9+225]=1170 (ii) aligned ) Now, after combined the -5 and 13 with the observations, the new variance ( aligned & = x₁^2+x₂^2+x₃^2+x₄^2+x₅^2+25+169 7 - ( 75-5+13 7 )^2 & = 1170+25+169 7 - ( 83 7 )^2= 9548-6889 49 = 2659 49 aligned ) Hence, option (d) is correct.