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:
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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 #