Question

The lengths of nails produced in a factory are normally distributed with a mean of 3.34 centimeters and a standard deviation of 0.07 centimeters. Find the two lengths that separate the top 3% and the bottom 3%. These lengths could serve as limits used to identify which nails should be rejected. Round your answer to the nearest hundredth, if necessary.

Answer 1

3.47 and 3.21

Explanation:

Let us assume the nails length be X

`X \sim N(3.34,0.07^2)`

Value let separated the top 3% is T and for bottom it would be B

`P(X < T)= 0.97`

Now converting, we get

`P(Z < (T-3.34)/(0.07))= 0.97`

Based on the normal standard tables, we get

`P(Z < 1.881)= 0.97`

Now compare these two above equations

`(T-3.34)/(0.07) = 1.881 \\\\ T = 1.881 * 0.07 + 3.34 \\\\ = 3.47`

So for top 3% it is 3.47

Now for bottom we applied the same method as shown above

`P(Z < (B-3.34)/(0.07))= 0.03`

Based on the normal standard tables, we get

`P(Z < -1.881)= 0.03`

Now compare these two above equations

`(B-3.34)/(0.07) = -1.881`

`= -1.881 * 0.07 + 3.34 \\\\ = 3.21`

hence, for bottom it would be 3.21