Assume you have a Normal Distribution with a **mean of 7.5** and a **standard deviation of 3.** - What is the probability that X is greater than 8.5?

$\displaystyle P(X > 8,5)$

$\displaystyle = 1 - P(X < 8,5)$

$\displaystyle = 1 - P( \frac{X - 7,5}{3} < \frac{8,5 - 7,5}{3} )$

$\displaystyle = 1 - P(Z < \frac{1}{3})$

$\displaystyle = 1 - \Phi (\frac{1}{3})$

$\displaystyle = 1 - 0,6293$

$\displaystyle = 0,3707$

- What is the probability that X is greater than 6.5?.

$\displaystyle P(X > 6,5)$

$\displaystyle = 1 - P(X < 6,5)$

$\displaystyle = 1 - P( \frac{X - 7,5}{3} < \frac{6,5 - 7,5}{3} )$

$\displaystyle = 1 - P(Z < - \frac{1}{3})$

$\displaystyle = 1 - \Phi (- \frac{1}{3})$

$\displaystyle = 1 - 0,3707$

$\displaystyle = 0,6293$

- What is the probability that X is greater than 3 but less than 5.5?

$\displaystyle P(3 < X < 5,5)$

$\displaystyle = P( \frac{3 - 7,5}{3} < \frac{X - 7,5}{3} < \frac{5,5 - 7,5}{3} )$

$\displaystyle = P(- \frac{3}{2} < Z < - \frac{2}{3})$

$\displaystyle = \Phi (- \frac{2}{3}) - \Phi (- \frac{3}{2})$

$\displaystyle = 0,2514 - 0,0668$

$\displaystyle = 0,1846$

- What is the probability that X is less than 7.5?

$\displaystyle P(X < 7,5)$

$\displaystyle = P( \frac{X - 7,5}{3} < \frac{7,5 - 7,5}{3} )$

$\displaystyle = P(Z < 0)$

$\displaystyle = \Phi (0)$

$\displaystyle = 0,5000$