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

Step 1: Use the mean to link a and b.
The mean of \(a, b, 8, 5, 10\) is \(6\), so their sum is \(5 \times 6 = 30\). This gives \(a + b + 8 + 5 + 10 = 30\), so \(a + b = 7\). Call this equation (1).

Step 2: Write the variance formula.
Variance is the average of squared deviations from the mean: \[ 6.80 = \frac{(a-6)^2 + (b-6)^2 + (8-6)^2 + (5-6)^2 + (10-6)^2}{5} \]

Step 3: Simplify the known terms.
\((8-6)^2 + (5-6)^2 + (10-6)^2 = 4 + 1 + 16 = 21\). Multiplying both sides by \(5\) gives \(34 = (a-6)^2 + (b-6)^2 + 21\), so \((a-6)^2 + (b-6)^2 = 13\). Call this equation (2).

Step 4: Substitute a from (1).
From (1), \(a = 7 - b\). Put this into (2): \((1-b)^2 + (b-6)^2 = 13\). Expanding gives \(1 - 2b + b^2 + b^2 - 12b + 36 = 13\), so \(2b^2 - 14b + 24 = 0\), which reduces to \(b^2 - 7b + 12 = 0\).

Step 5: Solve the quadratic.
Factoring gives \((b-3)(b-4) = 0\), so \(b = 3\) or \(b = 4\). If \(b = 4\), then \(a = 3\); if \(b = 3\), then \(a = 4\).

Step 6: Match with the options.
Option A gives mean \(6\) but a variance of \(11.6\), option B gives variance \(7.6\), and option D fails the mean check itself since \(2+4+8+5+10=29\). Only \(a=3, b=4\) satisfies both conditions exactly.

Final Answer:
The values are \(a=3\) and \(b=4\). \[ \boxed{a = 3, \ b = 4} \]