Question

Assume that X is normally distributed with a mean of 15 and a standard deviation of 2. Determine the value for x that solves:

P(X>x) = 0.5.

P(X < 13).

P(13 < X < 17).

Answer 1

a)

P(X > x) = 0.5

Or, P((X - )/ > (x - )/) = 0.5

Or, P(Z > (x - 15)/2) = 0.5

Or, P(Z < (x - 15)/2) = 0.5

Or, (x - 15)/2 = 0

Or, x = 0 * 2 + 15

Or, x = 15

b) P(X < 13)

= P((X - )/ < (13 - )/)

= P(Z < (13 - 15)/2)

= P(Z < -1)

= 0.1587

c) P(13 < X < 17)

= P((13 - )/ < (X - )/ < (17 - )/)

= P((13 - 15)/2 < Z < (17 - 15)/2)

= P(-1 < Z < 1)

= P(Z < 1) - P(Z < -1)

= 0.8413 - 0.1587

= 0.6826