Question

The Rockwell hardness of certain metal pins is known to have a mean µ = 50 and a standard deviation σ = 4. (a) (5 points) Assume that the distribution of all such pin hardness measurements is known to be normal. If we randomly select 1 pin from the population, what is the probability that the hardness is less than 46?

Answer 1

The probability that a randomly selected metal pin has a hardness less than 46 is approximately 15.87%, calculated by finding the Z-score and corresponding probability from the Z-table.

Step-by-step explanation:

To find the probability that a randomly selected pin has a hardness of less than 46, we calculate the Z-score for 46 using the given mean (μ = 50) and standard deviation (σ = 4). This involves using the formula:

Z = (X - μ) / σ

Substituting the values gives:

Z = (46 - 50) / 4

Z = -1

The Z-score tells us how many standard deviations an element is from the mean. A Z-score of -1 indicates that the value is one standard deviation below the mean. Since the distribution is normal, we can use the Z-table to find the probability associated with a Z-score of -1, which is approximately 0.1587. This means there is a 15.87% chance that a randomly selected pin from the population will have a hardness less than 46.