Let's break down and solve each part of the question using the Empirical Rule (also known as the 68-95-99.7 rule). This rule applies to a normal (bell-shaped) distribution and gives approximate percentages of data that lie within 1, 2, and 3 standard deviations from the mean.

Given information:

• Mean (μ) = 47 ounces
• Standard deviation (σ) = 10 ounces

a) 68% of the widget weights lie between:

68% of the data lies within 1 standard deviation of the mean. This means:

• Lower bound = Mean - 1 standard deviation = 47 - 10 = 37 ounces
• Upper bound = Mean + 1 standard deviation = 47 + 10 = 57 ounces

Thus, 68% of the widget weights lie between 37 and 57 ounces.

b) What percentage of the widget weights lie between 27 and 57 ounces?

• 27 ounces is 2 standard deviations below the mean (47 - 2 * 10 = 27).
• 57 ounces is 1 standard deviation above the mean.

Using the Empirical Rule:

• 95% of the data lies within 2 standard deviations from the mean (between 27 and 67 ounces).
• Since 68% of the data lies between 37 and 57 ounces, the remaining 27% (from 27 to 37 ounces) must be split in half to get the portion from 27 to 37.
• Therefore, the percentage between 27 and 57 is 68% + 13.5% = 81.5%.

c) What percentage of the widget weights lie below 77 ounces?

• 77 ounces is 3 standard deviations above the mean (47 + 3 * 10 = 77).
• According to the Empirical Rule, 99.7% of the data lies within 3 standard deviations, so half of the remaining 0.3% (since the question asks for "below" 77) must be subtracted from 100%.
• Thus, the percentage of weights below 77 ounces is approximately 99.85%.

Would you like any further details or clarification on these calculations?

Here are 5 related questions:

1. How would the answers change if the standard deviation was 5 ounces instead of 10?
2. What percentage of the weights lie above 77 ounces?
3. How does the Empirical Rule differ from using z-scores for precise calculations?
4. If the mean widget weight was increased to 50 ounces, what would be the new range for 68% of the weights?
5. How could you estimate the percentage of widget weights lying between 37 and 67 ounces?

Tip: The Empirical Rule is a quick approximation tool, but for exact probabilities, especially for non-standard intervals, you should use the z-score formula and a normal distribution table.