statistics

Question

the numbers a,b,8,5,7 have a mean of 6 and a variance of 2. find the values of a and b, if a>b.

Answer 1

As the mean is 6, and there are 5 values,

(a+b+8+5+7)/5 = 6

a+b+8+5+7 = 30

a + b = 30 - 8 - 5 - 7

a + b = 10

 

The variance shortcut formula is (the sum of xi ^ 2 - n x-bar ^ 2)/(n - 1)

As the mean is 6,

a^2+b^2 + 8^2 +5^2+7^2 - 5 * 6^2)/(5 - 1) = 2

(a^2 + b ^ 2 + 64 + 25 + 49 - 5 * 36)/4 = 2

a^2 + b^2 - 42 = 8

a^2 + b^2 = 50

 

Now, we can solve

a^2+b^2 = 50

a+b = 10

We know this has solution a=b=5  (We could solve by letting

b = 10 - a

a^2 + (10 - a)^2 = 50

2a^2 + 100 - 20a = 50

2a^2 - 20a + 50 = 0

a^2 - 10a + 25 = 0

(a - 5)^2 = 0

a = 5; thus, b = 5