Assume that X is normally distributed with a mean of 20 and a standard deviation of 2. Determine the following. (a) P (X 24) (b) P (X 18) (c) P (18 X 22) (d) P (14 X 26) (e) P (16 X 20) (f) P (20 X 26)

a) P (X < 24) = 0.9772

b) P (X > 18) = 0.8413

c) P (14 < X < 26) = 0.9973

d) P (14 < X < 26) = 0.9973

e) P (16 < X < 20) = 0.4772

f) P (20 < X < 26) = 0.4987

Step-by-step explanation:

Given:

- Mean of the distribution u = 20

- standard deviation sigma = 2

Find:

a. P (X < 24)

b. P (X > 18)

c. P (18 < X < 22)

d. P (14 < X < 26)

e. P (16 < X < 20)

f. P (20 < X < 26)

Solution:

- We will declare a random variable X that follows a normal distribution

X ~ N (20, 2)

- After defining our variable X follows a normal distribution. We can compute the probabilities as follows:

a) P (X < 24) ?

- Compute the Z-score value as follows:

Z = (24 - 20) / 2 = 2

- Now use the Z-score tables and look for z = 2:

P (X < 24) = P (Z < 2) = 0.9772

b) P (X > 18) ?

- Compute the Z-score values as follows:

Z = (18 - 20) / 2 = - 1

- Now use the Z-score tables and look for Z = - 1:

P (X > 18) = P (Z > - 1) = 0.8413

c) P (18 < X < 22) ?

- Compute the Z-score values as follows:

Z = (18 - 20) / 2 = - 1

Z = (22 - 20) / 2 = 1

- Now use the Z-score tables and look for z = - 1 and z = 1:

P (18 < X < 22) = P (-1 < Z < 1) = 0.6827

d) P (14 < X < 26) ?

- Compute the Z-score values as follows:

Z = (14 - 20) / 2 = - 3

Z = (26 - 20) / 2 = 3

- Now use the Z-score tables and look for z = - 3 and z = 3:

P (14 < X < 26) = P (-3 < Z < 3) = 0.9973

e) P (16 < X < 20) ?

- Compute the Z-score values as follows:

Z = (16 - 20) / 2 = - 2

Z = (20 - 20) / 2 = 0

- Now use the Z-score tables and look for z = - 2 and z = 0:

P (16 < X < 20) = P (-2 < Z < 0) = 0.4772

f) P (20 < X < 26) ?

- Compute the Z-score values as follows:

Z = (26 - 20) / 2 = 3

Z = (20 - 20) / 2 = 0

- Now use the Z-score tables and look for z = 0 and z = 3:

P (20 < X < 26) = P (0 < Z < 3) = 0.4987