The weights of certain machine components are normally distributed with a mean of 9.75g and a standard deviation of 0.08g. Find the weights that separate the top 5% and the bottom 5%.

9.62 and 9.88 grams, each of which are aboug 1.5 sigma from the mean.

To find the weights that separate the top 5% and the bottom 5%, we need to calculate the z-scores associated with these percentiles and then use those z-scores to find the corresponding weights.

The z-score formula is given by: Z = (X - μ) / σ

Where:
- Z is the z-score
- X is the value we want to find the z-score for
- μ is the mean of the distribution
- σ is the standard deviation of the distribution

First, let's find the z-score for the top 5%.

To find the z-score for the top 5%:
1. Subtract the desired area (percentage) from 1. In this case, 1 - 0.05 = 0.95.
2. Look up the z-score associated with this cumulative probability (0.95) using a standard normal distribution table or a calculator. The z-score is approximately 1.645.

Now, we can calculate the weight corresponding to this z-score using the formula:

X = Z * σ + μ

X = 1.645 * 0.08 + 9.75
X ≈ 9.885 g

Therefore, the weight that separates the top 5% is approximately 9.885g.

To find the weight that separates the bottom 5%, we use the same approach but with the desired area (percentage) of 5% instead.

The z-score for the bottom 5%:
1. Subtract the desired area (percentage) from 0.5. In this case, 0.5 - 0.05 = 0.45.
2. Look up the z-score associated with this cumulative probability (0.45) using a standard normal distribution table or a calculator. The z-score is approximately -1.645 (since we're looking for the bottom 5%).

Now, we can calculate the weight corresponding to this z-score using the formula:

X = Z * σ + μ

X = -1.645 * 0.08 + 9.75
X ≈ 9.615 g

Therefore, the weight that separates the bottom 5% is approximately 9.615g.