Answer:
(a) 259.5 days
(b) 284 days
Step-by-step explanation:
Let X represent the pregnancy length in days.
It is provided that [tex]X\sim N(270,9^{2})[/tex].
(a)
Let a represent the minimum pregnancy length that can be in the top 12% of pregnancy lengths.
Then,
P (X < a) = 0.12
⇒ P (Z < z) = 0.12
The corresponding z-score is, z = -1.17.
*Use a z-table.
Compute the value of x as follows:
[tex]z=\frac{x-\mu}{\sigma}\\\\-1.17=\frac{a-270}{9}\\\\z=270-(9\times 1.17)\\\\z=259.47\\\\z\approx 259.5[/tex]
Thus, the minimum pregnancy length that can be in the top 12% of pregnancy lengths is 259.5 days.
(b)
Let b be the maximum pregnancy length that can be in the bottom 6% of pregnancy lengths.
Then,
P (X > b) = 0.06
⇒ P (X < b) = 0.94
⇒ P (Z < z) = 0.94
The corresponding z-score is, z = 1.56.
*Use a z-table.
Compute the value of x as follows:
[tex]z=\frac{x-\mu}{\sigma}\\\\1.56=\frac{b-270}{9}\\\\z=270+(9\times 1.56)\\\\z=284.04\\\\z\approx 284[/tex]
Thus, the maximum pregnancy length that can be in the bottom 6% of pregnancy lengths is 284 days.
