If the average of 10 observations a1, a2, a3 …., a10 is b, then which of the following will be the average of a1-11, a2-11, a3-11, ……, a10-11?

The average of a set of numbers is calculated by adding all the numbers together and then dividing by the count of numbers.

Given that the average of 10 observations a1, a2, a3 …., a10 is b, this can be written as:

(a1 + a2 + a3 + ... + a10) / 10 = b

Multiply both sides by 10 to get rid of the denominator:

a1 + a2 + a3 + ... + a10 = 10b

Now, if we subtract 11 from each observation, the new set becomes a1-11, a2-11, a3-11, ……, a10-11. The sum of these observations is:

(a1 - 11) + (a2 - 11) + (a3 - 11) + ... + (a10 - 11)

This can be rewritten as:

(a1 + a2 + a3 + ... + a10) - (11 + 11 + 11 + ... + 11)

We know from the previous step that (a1 + a2 + a3 + ... + a10) = 10b, and (11 + 11 + 11 + ... + 11) = 110, so we can substitute these values in:

10b - 110

To find the average of these new observations, we divide this sum by the count of numbers, which is 10:

(10b - 110) / 10

Simplify this to get the final answer:

b - 11

So, the average of the observations a1-11, a2-11, a3-11, ……, a10-11 is (b - 11).