Answer:
The two lengths that separate the top 6% and the bottom 6% of the lengths are 4.94cm and 5.10cm respectively.
Step-by-step explanation:
Let the two lengths that separate the top and the bottom 6% be 'a' and 'b'.
Lengths of nails produced in a factory are normally distributed with a mean of 5.02 cm
Standard deviation of 0.05 cm
The distribution would be symmetric about the mean i.e. X = 5.02.
So, the lengths 'a' and 'b' will be symmetric with the center as probability on each side of the center = 0.5.
So, P(X < a) = P(X > b) = 0.06.
So, using the standardized normal score, we can write:
P(X < a) =[tex]P(Z < \frac{a - 5.02}{0.05})[/tex] = 0.06
From the Z-table we get the value of Z for which cumulative probability is 0.06 as
P ( Z < -1.555 ) = 0.06
so,
[tex]\frac{a - 5.02}{0.05}[/tex] = -1.555
a = 4.94225
As the value 'b' is symmetric with 'a' about center, so
5.02 - a = b - 5.02
10.04 - a = b
since a = 4.94225,
10.04 - 4.94225 = b
b = 5.09775
