Question 922356
<pre>
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:
</pre>
the mean score on a business exam is 68
<pre>
So {{{mu=68}}}
</pre>
the probability of a score greater than 76 is 6.5%
<pre>
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 {{{mu=68}}} in

{{{z = (x-mu)/sigma}}} to find standard deviation {{{sigma}}}

But first we'll solve that for {{{sigma}}}

{{{z*sigma = x-mu}}}

{{{sigma = (x-mu)/z}}}

Now we'll substitute:

{{{sigma = (76-68)/1.51}}}

{{{sigma=5.284}}}
</pre>
What is the probability of a score between 55 and 76?
<pre>
That's easy, now that we have the standard deviation {{{sigma=5.284}}}.

We find that to be 0.9281

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We didn't even need this at all:
</pre>
the probability of a score between 55 and 60 is 2.5% 
<pre>
However that is inconsistent with {{{sigma=5.284}}}.

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</pre>