I have a tough one for everyone.

Scores on an endurance test for cardiac patients are normally
distributed with mean = 182 and standard deviation  = 24.
       a.     What is the probability a patient will score above 190?
       b.     What percentage of patients score below 170?
       c.     What score does a patient at the 75th percentile receive?

Hi scifinut69!!

We have a normal distributed ramdom variable X with mean Mu = 182 and
standard deviation StD = 24 .

The first step is to normalize the variable X:
Z = (X-Mu)/STD = (X-182)/24

We will use the following table for calculations:
"Normal Distribution Table"
http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/normaltable.html


a.     What is the probability a patient will score above 190?

If X > 190 ==> Z > (190-182)/24 = 0.33

Then:
P(X > 190) = P(Z > 0.333) = 0.5 - P(0 < Z =< 0.33)

From the table:
P(0 < Z =< 0.33) = 0.1293

We have:
P(X > 190) = 0.5 - 0.1293 = 0.3707

The probability a patient will score above 190 is 37.07%

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b.     What percentage of patients score below 170?

If If X < 170 ==> Z > (170-182)/24 = -0.5

Due the symmetry of the graph of the normal distribution we have that:

P(X < 170) = P(Z < -0.5) = P(Z > 0.5) = 0.5 - P(0< Z < 0.5)

From the table:
P(0< Z < 0.5) = 0.1915

Then:
P(X < 170) = 0.5 - 0.1915 = 0.3085

The percentage of patients score below 170 is 30.85%.

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c.     What score does a patient at the 75th percentile receive?

The score at the 75th percentile is the score at which the 75% of the values are
smaller than or equal to it.
If Y is such score then 

P(Z =< S) = 0.75

Because 0.75 > 0.5 then S > 0 and:

0.75 = P(Z =< S) = 0.5 + P(0 < Z =< S) 

with 

P(0 < Z =< S) = 0.25

Now go to the table and find the closer A for the area 0.2500, the
closer area is 0.2486, and it is correspond for the value A = 0.67

Then S = 0.67
To find X from this value:
X = 0.67 * StD + Mu = 0.67 * 24 + 182 = 198.08

(Note that the found value is an aproximation, a more accuracy result
is 198.18775 for S = 0.674485 ).

A patient at the 75th percentile receive a score of 198.08 aprox.

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For additional references see:
"USING THE STANDARDIZED NORMAL DISTRIBUTION TABLE":
http://myphliputil.pearsoncmg.com/student/bp_berenson_bbs_9/section6_1.pdf

"The Normal Distribution":
http://www.wmich.edu/cpmp/unitsamples/c3u5/C3U5_362-375.pdf

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I hope that this helps you. Feel free to request for a clarification
if it is needed.

Regards.
livioflores-ga

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Clarification of Answer by livioflores-ga on 23 Nov 2004 00:40 PST

For more accurate results you can use the following "electronic tables":
http://davidmlane.com/hyperstat/z_table.html