Question

The lengths of a particular​ animal's pregnancies are approximately normally​ distributed, with mean = 280 days and standard deviation = 20 days.

​(a) What proportion of pregnancies lasts more than 285 ​days? ​

(b) What proportion of pregnancies lasts between 245 and 295 ​days? ​

(c) What is the probability that a randomly selected pregnancy lasts no more than 255 ​days? ​

​(d) The probability of a​ "very preterm" baby is ? This event would/would not be unusual because the probability is less/greater than 0.05?

*** A​ "very preterm" baby is one whose gestation period is less than 235 days. Are very preterm babies​ unusual?

Answer 1

Solution :

Given that,

mean = = 280

standard deviation = = 20

a ) P (x > 285 )

= 1 - P (x < 285 )

= 1 - P ( x -  / ) < ( 285 - 280 / 20)

= 1 - P ( z < 5 / 20 )

= 1 - P ( z < 0.25)

Using z table

= 1 - 0.5987

= 0.4013

Probability = 0.4013

b ) P (245 < x < 295​ )

P ( 245 - 280 / 20) < ( x -  / ) < ( 295​ - 280 / 20)

P ( - 35 / 20< z < 15 / 20 )

P (-1.75 < z < 0.75 )

P ( z < 0.75 ) - P ( z < -1.75 )

Using z table

= 0.7734 - 0.0401

= 0.7333

Probability = 0.7333

c ) P (x > 255 )

= 1 - P (x < 255 )

= 1 - P ( x -  / ) < ( 255 - 280 / 20)

= 1 - P ( z <- 25 / 20 )

= 1 - P ( z < -1.25 )

Using z table

= 1 - 0.1056

= 0.8944

Probability = 0.8944

d ) P( x < 235 )

P ( x - / ) < ( 235- 280 / 20)

P ( z < -45 / 20 )

P ( z < -2.25)

=0.0122

Probability = 0.0122

This event  unusual because the probability is less than 0.05

0.05 < 0..0122