The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8. then a ratio of other two observations is

Options

A. `4:9`
B. `6:7`
C. `10:3`
D. `5:8`

Answer

A

Explanation

Let 1, 3, 8, x and y be the five observations.
Then, mean `bar(x)=(Sigma x_(i))/(n)`
`rArr bar(x) =(1+3+8+x+y)/(5)=5 " " `(given)
`rArr x+y=25-12=13`
`rArr x+y=13 " ...(i)" `
and variance `=sigma^(2)=(Sigma(x_(i)-bar(x))^(2))/(n)`
`=([(1-5)^(2)+(3-5)^(2)+(8-5)^(2)+(x-5)^(2)+(y-5)^(2)])/(5)=9.2` (given)
`rArr 16+4+9+(x^(2)-10x+25)+(y^(2)-10y+25)=46`
`rArr x^(2)+y^(2)-10(x+y)=46-79`
`rArr x^(2)+y^(2)-10xx13= -33 " "[ because x+y=13]`
`rArr x^(2)+y^(2)=97 " ...(ii) " `
Let `(y)/(x)=t`
`rArr y=xt`
Putting y = xt in Eq. (i), we get
`x(1+t)=13`
`rArr x^(2)(1+t)^(2)=169 " ...(iii)" `
Putting y = xt in Eq. (ii), we get
`x^(2)(1+t^(2))=97 " ...(iv)" `
Dividing Eq. (iii) by Eq. (iv), we get
`(x^(2)(1+t)^(2))/(x^(2)(1+t^(2)))=(169)/(97)`
`rArr97(t^(2)+2t+1)=169(1+t^(2))`
`rArr (169-97)t^(2)-194t+(169-97)=0`
`rArr 36t^(2)-97t+36=0`
`rArr (4t-9)(9t-4)=0`
`rArr t=(9)/(4)`
`or t=(4)/(9)`