50 average profit
no negative profit
more 50 Percent > 100
a) Is Mr. Jokey claim possible ?
Yes but just barely
If half of the n phones sold had exactly 100 profit and half of the n phones had a 0 dollar profit , then
the means equal 50 exactly. If any the phone had a greater profit, then the mean would be greater than 50.
mean = (n/2)*100 + 0 *(n/2) = average (50n + 0 ) / n = 50
if any of the 1/2 at 100 was higher or any of the (1/2) at 0 was higher, then the average would greater than 50.
so this is only possible if at least 100 means 1/2 of them were 100
and if all the others one were 0.
However, this border line case,
(1/2) the phones had a profit of 100 and (1/2) the phones had a profit of 0
The average would still be 50 and half the phone were at least 100.
Then, his claims could be true.
Example if n = 50 , and 25 had profit equal 100 and the other 25 had a profit of a 0.02 (2 cents )
mean = (25*100 + 25*0.02 ) / 50 = (2500.50)/50 = 50.01 average
so even a two cents profit average on remaining phones would lift the average just above 50 dollars.
The only case that works is (1/2) at 100 and (1/2) at 0 .
This meets the claims at the borderline of the claims
100 is at least 100
0 is not a loss.
and with exactly (1/2) at 100, the average can still be 50.
No, because than the borderline case could no longer happen.
with half the entries at 100 and half the entries at 0, then the standard deviation
would be greater than 25 .
assume we have the population and not just a sample
borderline case standard deviation =sqrt ( (n/2)*(100-50)^2 + (n/2)*(50-0)^2 / n ) = sqrt ( n *50^2/n) = sqrt (50^2) = 50
for the standard deviation to be lower than 50, then some of the
at the low end would have to have a profit greater than 0, but
then the average would be greater than 50 .
with n/2 phones sell with 100 profit and the average fifty
than, less say all the other phone were the average cost say 50 dollars
standard deviation is over = sqrt ( (n/2)*50^2/ n ) = 50/sqrt(2) = 50*sqrt(2) /2 = 25*sqrt(2)
standard deviation is greater than 25
so even with half the phone at 50 dollar profit above the average the lowest standard deviation possible
is 25*sqrt(2) which is greater than 25 if (1/2 the phones had a profit of 100 dollars and overall average was 50.