Question

1. Length (in days) of human pregnancies is a normal random variable (X) with mean 266, standard deviation 16.

a. The probability is 95% that a pregnancy will last between what 2 days? (Remember your empirical rule here)

b. What is the probability of a pregnancy lasting longer than 315 days?

2. What is the probability that a normal random variable will take a value that is less than 1.05 standard deviations above its mean? In other words, what is P(Z < 1.05)?

3. What is the probability that a normal random variable will take a value that is between 1.5 standard deviations below the mean and 2.5 standard deviations above the mean? In other words, what is P(−1.5 < Z < 2.5)?

4. What is the probability that a normal random variable will take a value that is more than 2.55 standard deviations above its mean? In other words, what is P(Z > 2.55)?

Answer 1

Part 1)

X ~ N ( µ = 266 , σ = 16 )
P ( a < X < b ) = 0.95
Dividing the area 0.95 in two parts we get 0.95/2 = 0.475
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.475
Area above the mean is b = 0.5 + 0.475
Looking for the probability 0.025 in standard normal table to calculate Z score = -1.96
Looking for the probability 0.975 in standard normal table to calculate Z score = 1.96
Z = ( X - µ ) / σ
-1.96 = ( X - 266 ) / 16
a = 234.64
1.96 = ( X - 266 ) / 16
b = 297.36
P ( 234.64 < X < 297.36 ) = 0.95

Part b)

X ~ N ( µ = 266 , σ = 16 )
P ( X > 315 ) = 1 - P ( X < 315 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 315 - 266 ) / 16
Z = 3.0625
P ( ( X - µ ) / σ ) > ( 315 - 266 ) / 16 )
P ( Z > 3.0625 )
P ( X > 315 ) = 1 - P ( Z < 3.0625 )
P ( X > 315 ) = 1 - 0.9989
P ( X > 315 ) = 0.0011

Part 2)

X ~ N ( µ = 266 , σ = 16 )
Z = ( X - µ ) / σ
1.05 = ( X - 266 ) / 16
X = 282.8
P ( ( X - µ ) / σ ) < ( 282.8 - 266 ) / 16 )
P ( X < 282.8 ) = P ( Z < 1.05 )
P ( X < 282.8 ) = 0.8531

part 3)

Z = ( X - µ ) / σ
Z = ( X - 266 ) / 16

-1.5 = ( X - 266 ) / 16
X1 = 242

2.5 = ( X - 266 ) / 16
X2 = 306
P ( -1.5 < Z < 2.5 )
P ( 242 < X < 306 ) = P ( Z < 2.5 ) - P ( Z < -1.5 )
P ( 242 < X < 306 ) = 0.9938 - 0.0668
P ( 242 < X < 306 ) = 0.9270

Part 4)

Z = ( X - µ ) / σ
2.55 = ( X - 266 ) / 16
Z = 306.8
P ( Z > 2.55 )
P ( X > 306.8 ) = 1 - P ( Z < 2.55 )
P ( X > 306.8 ) = 1 - 0.9946
P ( X > 306.8 ) = 0.0054