To solve this problem, we need to use the given mean and variance to find possible values of
a and
b that satisfy these conditions.
The mean
(µ) of the numbers
a,b,8,5,10 is given as 6 . The formula for the mean is:
µ=‌Here, the sum of the numbers is:
a+b+8+5+10=a+b+23Since we have 5 numbers, the mean is:
‌=6 By solving for
a+b :
a+b+23=30a+b=7Now we know that
a+b=7. Next, let's use the given variance, which is 6.80 . The formula for the variance
(σ2) is:
σ2=‌ First, compute the squared deviation of each number from the mean 6 :
(a−6)2,‌‌(b−6)2,‌‌(8−6)2=4,‌‌(5−6)2=1,‌‌(10−6)2=16
Summing these deviations, we get:
(a−6)2+(b−6)2+4+1+16Since the variance is given as 6.80 , we can use the variance formula:
‌| (a−6)2+(b−6)2+4+1+16 |
| 5 |
=6.80 Multiplying this equation by 5 :
(a−6)2+(b−6)2+21=34Therefore,
(a−6)2+(b−6)2=13 Now, we solve for each option to see which satisfies both equations:
Option A
‌a=1,‌‌b=6‌a+b=1+6=7‌(1−6)2+(6−6)2=25+0=25 This does not satisfy the variance equation.
Option B
‌a=0,‌‌b=7‌a+b=0+7=7‌(0−6)2+(7−6)2=36+1=37 This does not satisfy the variance equation.
Option C
‌a=3,‌‌b=4‌a+b=3+4=7‌(3−6)2+(4−6)2=9+4=13 This satisfies both the mean and variance equations.
Option D
‌a=5,‌‌b=2‌a+b=5+2=7‌(5−6)2+(2−6)2=1+16=17This does not satisfy the variance equation.
Therefore, from the given options, Option
Ca=3,b=4 is the correct pair that meets the given mean and variance conditions.