Assume the random variable X is normally distributed with mean μ = 50 and standard deviation σ = 7. What is the probability P (X > 42)?

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Answer 1 · 2017-02-28T21:42:41

$P(X>42) = 0.1271$

We must standardise the Random Variable $X$ with the Standardised Normal Distribution $Z$ Variable using the relationship:

$Z=(X-mu)/sigma$

And we will use Normal Distribution Tables of the function:

$Phi(z) = P(Z le z)$

And so we get:

$P(X>42) = P( Z > (42-50)/7 )$
$" " = P( Z > -8/7 )$
$" " = P( Z > -1.1429 )$

If we look at this graphically it is the shaded part of this Standardised Normal Distribution:
enter image source here

By symmetry of the Standardised Normal Distribution it is the same as this shaded part
enter image source here

So;

$P(X>42) = P( Z > -1.1429 )$
$" " = 1- P( Z < 1.1429 )$
$" " = 1-Phi(1.1429 )$
$" " = 1-0.8729 \ \ \ \ \$ (from tables)
$" " = 0.1271$