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

Feb 28, 2017

$P \left(X > 42\right) = 0.1271$

#### Explanation:

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

$Z = \frac{X - \mu}{\sigma}$

And we will use Normal Distribution Tables of the function:

$\Phi \left(z\right) = P \left(Z \le z\right)$

And so we get:

$P \left(X > 42\right) = P \left(Z > \frac{42 - 50}{7}\right)$
$\text{ } = P \left(Z > - \frac{8}{7}\right)$
$\text{ } = P \left(Z > - 1.1429\right)$

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

By symmetry of the Standardised Normal Distribution it is the same as this shaded part

So;

$P \left(X > 42\right) = P \left(Z > - 1.1429\right)$
$\text{ } = 1 - P \left(Z < 1.1429\right)$
$\text{ } = 1 - \Phi \left(1.1429\right)$
$\text{ } = 1 - 0.8729 \setminus \setminus \setminus \setminus \setminus$ (from tables)
$\text{ } = 0.1271$