Answer:
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
