The mean of five observations is 5 and their variance is 9.20.

Question

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 the ratio of the other two observations is:

Accepted Answer

Mean $=5 \implies \sum x_i=25 \implies$ with $1+3+8=12$: $x_4+x_5=13$

Variance $=9.2 \implies \sum x_i^2 = 5(25+9.2)=171$

$1+9+64+x_4^2+x_5^2=171 \implies x_4^2+x_5^2=97$

$(x_4+x_5)^2=169=97+2x_4x_5 \implies x_4x_5=36$

Solving $x_4+x_5=13, x_4x_5=36$: $x_4=4,x_5=9$ or $x_4=9,x_5=4$.

$$x_4:x_5 = 4:9 \text{ or } 9:4$$

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