`"Mean"=10rArr(overset(20)underset(i=1)Sigmax_(i))/(20)=10`
`rArr" "overset(20)underset(i=1)Sigmax_(i)=200.`
`SD=2rArrsigma^(2)=4`
`rArr" "(overset(20)underset(i=1)Sigmax_(i)^(2))/(20)-(10)^(2)=4`
`rArr" "overset(20)underset(i=1)Sigmax_(i)^(2)=2080.`
Thus, incorrect `(overset(20)underset(i=1)Sigmax_(i)^(2))=200 and" incorrect "(overset(20)underset(i=1)Sigmax_(i)^(2))=2080.`
CASE(i) When the wrong item is omitted
On omiting 8, we are left with 19 observations.
`therefore" correct "(overset(19)underset(i=1)Sigmax_(i))=" incorrect "(overset(20)underset(i=1)Sigma)x_(i))-8`
`=(200-8)=192.`
Thus, correct `(overset(19)underset(i=1)Sigmax_(i))=192`
`therefore" correct mean"=(192)/(19)=10.105." ...(i)"`
`"Also, correct "(overset(19)underset(i=1)Sigmax_(i)^(2))=" incorrect "(overset(20)underset(i=1)Sigmax_(i)^(2))-64`
`=(2080-64)=2016.`
`therefore" correct variance "(1)/(19)("correct "overset(19)underset(i=1)Sigmax_(i)^(2))-("correct mean")^(2)`
`=((1)/(19)xx2016)-((192)/(19))^(2)=((2016)/(19)-(36864)/(361))`
`=((38304-36864))/(361)=(1440)/(361).`
`therefore" correct "SD=sqrt((1440)/(361))=(12sqrt(10))/(19)=((12xx3.162))/(19)`
`=(37.944)/(19)=1.997.`
`therefore" mean"=10.105 and SD=1.997.`
CASE(ii) When incorrect observation 8 is replaced by 12.
`"Incorrect"(overset(20)underset(i=1)Sigmax_(i))=200`
`rArr" correct"(overset(20)underset(i=1)Sigmax_(i))=(200-8+12)=204`
`rArr" correct mean "=(204)/(20)=10.2.`
Also, incorrect `(overset(20)underset(i=1)Sigmax_(i))=2080`
`rArr" correct"(overset(20)underset(i=1)Sigmax_(i)^(2))=(2080-8^(2)+12^(2))=2160`
`rArr" correct variance "=("correct"(overset(20)underset(i=1)Sigmax_(i)^(2)))/(20)-("correct mean")^(2)`
`={(2160)/(20)-(10.2)^(2)}=(108-104.04)=3.96.`
`rArr" correct "SD=sqrt(3.96)=1.989`
Thus, in this case, we have mean =10.2 and SD=1.989.
