The arithmetic mean of the observations 10,8,5,a,b is 6 and their variance 6.8. Then ab=
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6
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4
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3
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12
Mean = (10+8+5+a+b)/5 = 6 => 23+a+b = 30 => a+b=7. Variance = (sum(x^2)/5) - mean^2 = 6.8. (100+64+25+a^2+b^2)/5 - 36 = 6.8. (189+a^2+b^2)/5 = 42.8. 189+a^2+b^2 = 214. a^2+b^2 = 25. Since a+b=7 and a^2+b^2=25, (a+b)^2 = a^2+b^2+2ab => 49 = 25+2ab => 2ab=24 => ab=12.
Using the arithmetic mean formula, the sum of the observations is 10 + 8 + 5 + a + b = 6 times 5, which simplifies to 23 + a + b = 30 and yields a + b = 7. Applying the variance formula, the sum of squared deviations from the mean 6 equals 5 times 6.8 = 34. This sets up the equation 16 + 4 + 1 + (a - 6) squared + (b - 6) squared = 34, resulting in (a - 6) squared + (b - 6) squared = 13. Substituting b = 7 - a into the equation and solving the resulting quadratic yields the roots 3 and 4, meaning the values of a and b are 3 and 4. Therefore, the product ab equals 12.