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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 isA. `4:9`B. `6:7`C. `10:3`D. `5:8` |
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Answer» Correct Answer - A 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)` |
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