Question

The time required to assemble an electronic component is normally distributed with a mean and a standard deviation of 16 minutes and 4 minutes, respectively. It is unusual for the assembly time to be above 24 minutes or below 6 minutes. What proportion of assembly times fall in these unusual categories?

Answer 1

To find the proportion of assembly times that fall in the unusual categories (above 24 minutes or below 6 minutes) in a normally distributed process, you can use the Z-score formula. The Z-score tells you how many standard deviations a value is from the mean.

First, calculate the Z-scores for 24 minutes and 6 minutes:

For 24 minutes:
Z = (X - μ) / σ
Z = (24 - 16) / 4 = 2

For 6 minutes:
Z = (6 - 16) / 4 = -2.5

Now, you need to find the probabilities associated with these Z-scores. You can use a standard normal distribution table or calculator to find these probabilities.

For Z = 2, the probability is approximately 0.9772.
For Z = -2.5, the probability is approximately 0.0062.

Now, you need to find the proportion of assembly times in these unusual categories:

Proportion above 24 minutes = 1 - Probability(Z ≤ 2) = 1 - 0.9772 ≈ 0.0228

Proportion below 6 minutes = Probability(Z ≤ -2.5) ≈ 0.0062

So, the proportion of assembly times that fall in these unusual categories is approximately 0.0228 (above 24 minutes) and 0.0062 (below 6 minutes).