Question

The mean incubation time of fertilized eggs is 20 days. Suppose the incubation times are approximately...

The mean incubation time of fertilized eggs is 20 days. Suppose the incubation times are approximately normally distributed with a standard deviation of 1 day. ​(a) Determine the 14th percentile for incubation times. ​(b) Determine the incubation times that make up the middle 95​% of fertilized eggs.

Part a)
X ~ N ( µ = 20 , σ = 1 )
P ( X < x ) = 14% = 0.14
To find the value of x
Looking for the probability 0.14 in standard normal table to calculate Z score = -1.0803
Z = ( X - µ ) / σ
-1.0803 = ( X - 20 ) / 1
X = 18.9197
P ( X < 18.9197 ) = 0.14

Part b)
X ~ N ( µ = 20 , σ = 1 )
P ( a < X < b ) = 0.95
Dividing the area 0.95 in two parts we get 0.95/2 = 0.475
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.475
Area above the mean is b = 0.5 + 0.475
Looking for the probability 0.025 in standard normal table to calculate Z score = -1.96
Looking for the probability 0.975 in standard normal table to calculate Z score = 1.96
Z = ( X - µ ) / σ
-1.96 = ( X - 20 ) / 1
a = 18.04
1.96 = ( X - 20 ) / 1
b = 21.96

P ( 18.04 < X < 21.96 ) = 0.95

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