Something is wrong with this problem. The two pieces of
information given are inconsistent. We only need that
the mean is 68 and P(x > 76) = 0.065 to find the
standard deviation, from which we can easily find
P(55 < x < 76).
However using that standard deviation, We get that
P(55 < x < 60) = 0.058, not the 0.025 as is given.
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Here, I'll show this:
the mean score on a business exam is 68
So
the probability of a score greater than 76 is 6.5%
Therefore the z-score of x=76
We use a table or use a TI-83 or 84, to find the z-score
such that the area to the right of it is 0.065. We find
that as z = 1.51. That is,
P(z > 1.51) = 0.065
So want to substitute z = 1.51 x = 76 and in
to find standard deviation
But first we'll solve that for
Now we'll substitute:
What is the probability of a score between 55 and 76?
That's easy, now that we have the standard deviation .
We find that to be 0.9281
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We didn't even need this at all:
the probability of a score between 55 and 60 is 2.5%
However that is inconsistent with .
Because using the standard deviation that we calculated, that
P(55 < z < 60) = 0.058
So the two given pieces of information are inconsistent.
Edwin