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If a certain normal distribution of $X$, the probability is $0.5$ that $X$ is less than $500$ and $0.0227$ that $X$ is greater than $650$. What is the standard deviation of $X$?

How to compute S.D, of above?
in Probability by Active (4.7k points)
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I approached this way:

Let, Mean  = m

       Standard Deviation  = s

From the first data we get the Mean value.

P(X < 500) = P( (Z*s + m) < 500) = 0.5

Since, graph is symmetrical on both sides and each side probability = 0.5. The first data has probability = 0.5. This means that it is transformed into P(Z < 0).

Therefore, (500 - m)/s = 0. This gives mean value = 500

Now, second data states that: P(X > 650) = 0.0227

=> P( Z*s + 500 > 650) = 0.0227

=> P( Z*s > 150) = 0.0227

=>P(Z > (150/s) ) = 0.0227

=> 0.5 - P(0 < Z < (150/s) ) = 0.0227

=> P(0 < Z < (150/s)) = 0.4773

Now consulting the Table: http://socr.ucla.edu/Applets.dir/Z-table.html

I found out that P(0 < Z < 2) = 0.4772

Thus, I concluded (150/s) = 2

s = 150/2 = 75

The Standard Deviation = 75

by (163 points)

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