Question

The mean of the numbers \(a\), \(b\), \(8\), \(5\), \(10\) is \(6\) and the variance is \(6.80\). Which one of the following gives possible values of \(a\) and \(b\)?

• A. \(a = 0, b = 7\)
• B. \(a = 5, b = 2\)
• C. \(a = 3, b = 4\)
• D. \(a = 2, b = 4\)

Hint:

Use the mean to write \(a+b=7\), then plug this into the variance formula to get a quadratic in \(b\).

Answer 1

The Correct Option is C

Instead of solving the equations, test every option directly against both the mean and the variance conditions given for $a, b, 8, 5, 10$.

1. Option A ($a=0, b=7$): the sum is $0+7+8+5+10=30$, so the mean is $6$, which checks out. But the variance works out to $\frac{36+1+4+1+16}{5} = 11.6$, far from $6.80$, so this is wrong.
2. Option B ($a=5, b=2$): the sum is $5+2+8+5+10=30$, mean $6$ again checks out. The variance is $\frac{1+16+4+1+16}{5} = 7.6$, close but not equal to $6.80$, so this is wrong.
3. Option C ($a=3, b=4$): the sum is $3+4+8+5+10=30$, mean $6$ checks out. The variance is $\frac{9+4+4+1+16}{5} = \frac{34}{5} = 6.80$, matching exactly.
4. Option D ($a=2, b=4$): the sum is $2+4+8+5+10=29$, giving a mean of $5.8$, which already fails the mean condition, so this is wrong.

Only option C keeps both the mean at $6$ and the variance at $6.80$, so it is the answer.

Let's summarize:

• Check the mean first since it quickly rules out any option with the wrong sum.
• Then verify the variance on the remaining options to confirm the exact match.

This plug-in method avoids solving a quadratic and reaches the same result.