Question

The mean incubation time of fertilized eggs is 23 days. Suppose the incubation times are approximately normally distributed with a standard deviation of 1 day. ​(a) Determine the 11th percentile for incubation times. ​(b) Determine the incubation times that make up the middle 39​%. （2 numbers）

Answer 1

Part a)

X ~ N ( µ = 23 , σ = 1 )
P ( X < x ) = 11% = 0.11
To find the value of x
Looking for the probability 0.11 in standard normal table to calculate Z score = -1.23
Z = ( X - µ ) / σ
-1.2265 = ( X - 23 ) / 1
X = 21.77
P ( X < 21.77 ) = 0.11

Part b)

X ~ N ( µ = 23 , σ = 1 )
P ( a < X < b ) = 0.39
Dividing the area 0.39 in two parts we get 0.39/2 = 0.195
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.195
Area above the mean is b = 0.5 + 0.195
Looking for the probability 0.305 in standard normal table to calculate Z score = -0.51
Looking for the probability 0.695 in standard normal table to calculate Z score = 0.51
Z = ( X - µ ) / σ
-0.5101 = ( X - 23 ) / 1
a = 22.49
0.5101 = ( X - 23 ) / 1
b = 23.51

P ( 22.49 < X < 23.51 ) = 0.39