We use the definitions of mean and variance to set up two equations in a and b, then solve the system to find which option satisfies both.
We are given five numbers: a,b,8,5,10. Their mean is 6 and their variance is 6.80. We need to find which pair (a,b) from the options fits both conditions.
Concept & Intuition
The mean gives us a linear equation: the sum of all numbers equals 5×6=30.
The variance gives us a quadratic condition: the average of the squared deviations from the mean equals 6.80.
Together, these two equations will restrict a and b to a specific pair — we just check which option satisfies both.
Step-by-step solution
- Use the mean to get a relation between a and b.
5a+b+8+5+10=6
Multiply both sides by 5:
a+b+23=30⇒a+b=7
- Use the variance formula.
Variance σ2 is the average of (xi−mean)2. Here mean = 6, so:
σ2=5(a−6)2+(b−6)2+(8−6)2+(5−6)2+(10−6)2=6.80
Compute the known terms:
(8−6)2=4,(5−6)2=1,(10−6)2=16
Sum of known squares = 4+1+16=21.
So:
5(a−6)2+(b−6)2+21=6.80
Multiply by 5:
(a−6)2+(b−6)2+21=34
(a−6)2+(b−6)2=13
- Solve the system.
From step 1: b=7−a. Substitute into the squared equation:
(a−6)2+(7−a−6)2=13
Simplify the second term: 7−a−6=1−a, so:
(a−6)2+(1−a)2=13
Expand:
(a2−12a+36)+(1−2a+a2)=13
2a2−14a+37=13
2a2−14a+24=0
Divide by 2:
a2−7a+12=0
Factor:
(a−3)(a−4)=0
So a=3 or a=4.
If a=3, then b=7−3=4.
If a=4, then b=7−4=3.
- Match with options.
The pair (a,b)=(3,4) appears as option (C). The pair (4,3) is not listed, so only (C) works.
A common mistake is to forget that variance uses the mean of squared deviations, not the sum. Always divide by the number of data points.
Once you have a+b=7, you can quickly test each option: only (C) gives 3+4=7. Then just verify the variance condition to be sure.
✓Final answer
The correct option is (C).
ANSWER: C