Answer:
Two diameters that separate the top 7% and the bottom 7% are 5.15 and 5.03 respectively.
Step-by-step explanation:
We are given that the lengths of nails produced in a factory are normally distributed with a mean of 5.09 centimeters and a standard deviation of 0.04 centimeters.
Let X = lengths of nails produced in a factory
So, X ~ N(
)
The z score probability distribution is given by;
Z =
~ N(0,1)
where,
= population mean
= standard deviation
Now, we have to find the two lengths that separate the top 7% and the bottom 7%.
- Firstly, Probability that the diameter separate the top 7% is given by;
P(X > x) = 0.07
P(
>
) = 0.07
P(Z >
) = 0.07
So, the critical value of x in z table which separate the top 9% is given as 1.4996, which means;
= 1.4996
![x-5.09 = 0.04 \times 1.4996](https://tex.z-dn.net/?f=x-5.09%20%3D%200.04%20%5Ctimes%201.4996)
= 5.09 + 0.059984 = 5.15
- Secondly, Probability that the diameter separate the bottom 7% is given by;
P(X < x) = 0.07
P(
<
) = 0.07
P(Z <
) = 0.07
So, the critical value of x in z table which separate the bottom 9% is given as -1.4996, which means;
= -1.4996
![x-5.09 = 0.04 \times (-1.4996)](https://tex.z-dn.net/?f=x-5.09%20%3D%200.04%20%5Ctimes%20%28-1.4996%29)
= 5.09 - 0.059984 = 5.03
Therefore, the two diameters that separate the top 7% and the bottom 7% are 5.15 and 5.03 respectively.