Question

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 262 days and standard deviation 20 days.

Use a normal z-table to answer all questions below. If not,your answers may be marked as incorrect due to rounding issues.

(a) What proportion of pregnancies last less than 270 days (about 9 months)?
(Use 4 decimal places)

(b) What proportion of pregnancies last between 240 and 270 days (roughly between 8 months and 9 months)?
(Use 4 decimal places)

(c) The longest 20% of pregnancies last  days or longer. (Round to an integer.)

(d) What are the quartiles of the distribution of lengths of human pregnancies?

Q1 = __ days (Round to an integer.)
Q3 = ___ days (Round to an integer.)

Answer 1

Part a)
X ~ N ( µ = 262 , σ = 20 )
P ( X < 270 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 270 - 262 ) / 20
Z = 0.4
P ( ( X - µ ) / σ ) < ( 270 - 262 ) / 20 )
P ( X < 270 ) = P ( Z < 0.4 )
P ( X < 270 ) = 0.6554

Part b)
X ~ N ( µ = 262 , σ = 20 )
P ( 240 < X < 270 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 240 - 262 ) / 20
Z = -1.1
Z = ( 270 - 262 ) / 20
Z = 0.4
P ( -1.1 < Z < 0.4 )
P ( 240 < X < 270 ) = P ( Z < 0.4 ) - P ( Z < -1.1 )
P ( 240 < X < 270 ) = 0.6554 - 0.1357
P ( 240 < X < 270 ) = 0.5198

Part c)
X ~ N ( µ = 262 , σ = 20 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.2 = 0.8
To find the value of x
Looking for the probability 0.8 in standard normal table to calculate critical value Z = 0.8416
Z = ( X - µ ) / σ
0.8416 = ( X - 262 ) / 20
X = 278.832
P ( X > 278.832 ) = 0.2

Part d)

Q1 = 25%
X ~ N ( µ = 262 , σ = 20 )
P ( X < x ) = 25% = 0.25
To find the value of x
Looking for the probability 0.25 in standard normal table to calculate critical value Z = -0.6745
Z = ( X - µ ) / σ
-0.6745 = ( X - 262 ) / 20
X = 248.51
P ( X < 248.51 ) = 0.25

Q3 = 75%
X ~ N ( µ = 262 , σ = 20 )
P ( X < x ) = 75% = 0.75
To find the value of x
Looking for the probability 0.75 in standard normal table to calculate critical value Z = 0.6745
Z = ( X - µ ) / σ
0.6745 = ( X - 262 ) / 20
X = 275.49
P ( X < 275.49 ) = 0.75