Question

The lengths of a particular​ animal's pregnancies are approximately normally​ distributed, with mean μ=252 days and standard deviation σ=20 days.

​(a) What proportion of pregnancies lasts more than 267 days?

​(b) What proportion of pregnancies lasts between 227 and 257 ​days?

​(c) What is the probability that a randomly selected pregnancy lasts no more than 237 ​days?

​(d) A​ "very preterm" baby is one whose gestation period is less than 202 days. Are very preterm babies​ unusual?

Answer 1

Part a)

X ~ N ( µ = 252 , σ = 20 )
P ( X > 267 ) = 1 - P ( X < 267 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 267 - 252 ) / 20
Z = 0.75
P ( ( X - µ ) / σ ) > ( 267 - 252 ) / 20 )
P ( Z > 0.75 )
P ( X > 267 ) = 1 - P ( Z < 0.75 )
P ( X > 267 ) = 1 - 0.7734
P ( X > 267 ) = 0.2266

Part b)

X ~ N ( µ = 252 , σ = 20 )
P ( 227 < X < 257 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 227 - 252 ) / 20
Z = -1.25
Z = ( 257 - 252 ) / 20
Z = 0.25
P ( -1.25 < Z < 0.25 )
P ( 227 < X < 257 ) = P ( Z < 0.25 ) - P ( Z < -1.25 )
P ( 227 < X < 257 ) = 0.5987 - 0.1056
P ( 227 < X < 257 ) = 0.4931

Part c)

X ~ N ( µ = 252 , σ = 20 )
P ( X < 237 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 237 - 252 ) / 20
Z = -0.75
P ( ( X - µ ) / σ ) < ( 237 - 252 ) / 20 )
P ( X < 237 ) = P ( Z < -0.75 )
P ( X < 237 ) = 0.2266

Part d)

X ~ N ( µ = 252 , σ = 20 )
P ( X < 202 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 202 - 252 ) / 20
Z = -2.5
P ( ( X - µ ) / σ ) < ( 202 - 252 ) / 20 )
P ( X < 202 ) = P ( Z < -2.5 )
P ( X < 202 ) = 0.0062

Yes,   preterm babies​ unusual because the probability is less than 5% i.e 0.05.