Question

the length of human pregnancies from conception to birth varies according to an approximately normal distribution with a mean of 266 days and a standard deviation of 16 days.

a) what percent of pregnancies will last fewer than 215 days?

b) what is the probability that a randomly selected pregnancy lasts between 250 and 280 days?

c) 7.5% of all pregnancies last longer than how many days?

d) the central of 70% of all pregnancies lengths fall between what two values?

Answer 1

Solution :

Given that ,

a) P(x < 215)

= P[(x - ) / < (215 - 266) / 16 ]

= P(z < -3.1875 )

Using z table,

= 0.0007

b) P(250 < x < 280) = P[(250 - 266)/16 ) < (x - ) /  < (280 - 266) / 16) ]

= P(-1.0 < z < 0.875 )

= P(z < 0.875 ) - P(z < -1.0 )

Using z table,

= 0.8092 - 0.1587

= 0.6505

c) Using standard normal table,

P(Z > z) = 7.5%

= 1 - P(Z < z) = 0.075  

= P(Z < z) = 1 - 0.075

= P(Z < z ) = 0.925

= P(Z < 1.44) = 0.925

z = 1.44

Using z-score formula,

x = z * +

x = 1.44 * 16 + 266

x = 289.04

= 289 days

d) Using standard normal table,

P( -z < Z < z) = 70%

= P(Z < z) - P(Z <-z ) = 0.70

= 2P(Z < z) - 1 = 0.70

= 2P(Z < z) = 1 + 0.70

= P(Z < z) = 1.70 / 2

= P(Z < z) = 0.85

= P(Z < 1.036) = 0.85

= z  ± 1.036

Using z-score formula,

x = z * +

x = -1.036 * 16 + 266

x = 249.42

= 249 days

Using z-score formula,

x = z * +

x = 1.036 * 16 + 266

x = 282.57

= 283 days

The central 70% are from 249 days to 283 days