Question

In a recent​ study, the distances of certain raptor nests to the nearest marshland were studied. The study found that these distances are normally distributed with mean 4.68 km and standard deviation 0.85 km. Let Y be the distance of a randomly selected nest to the nearest marshland. Complete parts​ (a) and​ (b).

a. Determine P (Y > 5). ​(Round to four decimal places as​ needed.)

b. Determine P( 3 ≤ Y ≤ 6).

Answer 1

SOLUTION:

From given data,

The study found that these distances are normally distributed with mean 4.68 km and standard deviation 0.85 km. Let Y be the distance of a randomly selected nest to the nearest marshland.

Where ,

mean = = 4.68 km

standard deviation = = 0.85 km

z= x- /  

a. Determine P (Y > 5).

P (Y > 5) =P(Z> 5 - 4.68 /0.85)

= P(Z> 0.32/0.85)

= P(Z>0.376)

= 1-P (Z< 0.376)
= 1−0.6443

=0.3557

b. Determine P( 3 ≤ Y ≤ 6).

P(3≤ Y ≤ 6) =

= [((3 - 4.68) / 0.85) < (( X - ) / ) <  (( 6 - 4.68) / 0.85)]

= [-1.68 /0.85  < Z <  1.32 / 0.85]

= [-1.97 < Z <1.55]

=( Z < 1.55) - ( Z < - 1.97)

= 0.9394 - 0.0244

= 0.915