Question

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 178.9-cm and a standard deviation of 1-cm.

Find P84, which is the length separating the shortest 84% rods from the longest 16%.

P84 = ____ -cm

Answer 1

To find P84, which separates the shortest 84% of the steel rods from the longest 16%, we locate the z-score corresponding to the cumulative area of 0.84 and use the standard normal distribution to calculate the rod length at the 84th percentile, resulting in P84 = 179.9 cm.

Step-by-step explanation:

The student is asking to find the value of P84, which is the point in the normal distribution separating the shortest 84% of the steel rods from the longest 16%. Since the mean of the steel rods' lengths is 178.9 cm with a standard deviation of 1 cm, we can use the standard normal distribution (Z-distribution) to find this.

To find P84, we look for the z-score that corresponds to the cumulative area of 0.84 in the standard normal distribution table. The z-score that corresponds to 0.84 is approximately z = 1. After that, we can use the z-score formula (z = (x - mean)/standard deviation) to find the length of the steel rod at the 84th percentile (P84).

Using the formula, we get P84 = mean + z × standard deviation, which is 178.9 cm + 1 × 1 cm, resulting in P84 = 179.9 cm.