statistics question?

Question

the mean and standard deviation of a sample of wages in a group of firms are estimated at $ 10500 and 200 respectively. later, it is found that a wrong wage of $ 8000 of one person was included in the sample instead of the correct $7000.please can you be able to find
a) the correct mean wage
b)the correct standard deviation

Answer 1

Not without knowing how many wages were sampled.

If there were n wages sampled, then the correct mean is 10500 - 1000/n. The formula for the correct sample standard deviation is more complex, but it can be recovered if we know the number of elements in the sample.

For instance, suppose there were 100 wages sampled (so the true mean is $10490).
The formula for the sample standard deviation is
√(Σ(x - μ)^2 / (n-1))
so we know 40000(99) = Σ(x' - 10500)^2
where I use x' to indicate that these are the original values (in particular, one is $8000 which needs to be $7000 when we recalculate).
Now it's well known that Σ(i=1 to n) (x-μ)^2 = (Σ(i=1 to n) x^2) - n mu;^2
(since Σ(i=1 to n) (x-μ)^2 = Σ(i=1 to n) (x^2 - 2xμ + μ^2) = Σ(i=1 to n) (x^2) - 2μΣ(i=1 to n) x + n μ^2 = Σ(i=1 to n) (x^2) - 2μ(n μ) + n μ^2 = (Σ(i=1 to n) x^2) - n μ^2).

So we have 40000(99) = Σ(x')^2 - (100)10500^2
and so Σ(x')^2 = 11028960000.
So Σ(x^2) = 11028960000 - 8000^2 + 7000^2 = 11013960000. So the new sample standard deviation is
√((11013960000 - 100(10490)^2) / 99) = 317.

In actual fact, it would be impossible for a set of 100 wages with mean $10,500 and standard deviation $200 to have one member as low as $8,000 (which, be it noted, is 12.5 standard deviations below the mean. Are you sure it's not supposed to be $2000?) - a set of 100 wages with mean $10,500 and one element of $8,000 would have a standard deviation of 252.5 at the minimum, and that only if the other 99 values were all $10,525.25. (See also Chebyshev's Theorem.) To get a standard deviation of 200 would require 158 samples at the very least.

Answer 2

a) Let x be the number of items in the sample
Then
correct mean=(10500x -8000+7000)/x

=10500 -1000/x