Let's use the Empirical Rule to answer the questions based on the provided information:

Given:

• Mean (μ) = 60 ounces
• Standard deviation (σ) = 6 ounces

The Empirical Rule states that for a normal distribution:

• 68% of the data falls within 1 standard deviation of the mean (μ ± 1σ).
• 95% of the data falls within 2 standard deviations of the mean (μ ± 2σ).
• 99.7% of the data falls within 3 standard deviations of the mean (μ ± 3σ).

a) 95% of the widget weights lie between:

• 95% means within 2 standard deviations of the mean.
• Calculate:
  • Lower bound: $60 - 2 \times 6 = 60 - 12 = 48$
  • Upper bound: $60 + 2 \times 6 = 60 + 12 = 72$

Thus, 95% of the widget weights lie between 48 and 72 ounces.

b) What percentage of the widget weights lie between 54 and 72 ounces?

• To find the percentage, check how many standard deviations 54 and 72 are from the mean:

  • 54 is $60 - 6 = 1$ standard deviation below the mean (1σ below).
  • 72 is $60 + 12 = 2$ standard deviations above the mean (2σ above).

• From the Empirical Rule, we know:

  • From $-1σ$ to $1σ$, 68% of data lies.
  • From $-1σ$ to $+2σ$, 13.5% of the data lies (between 1σ and 2σ above).

So, the total percentage between 54 and 72 ounces is $68\% + 13.5\% = 81.5\%$.

c) What percentage of the widget weights lie below 78 ounces?

• 78 is $60 + 18 = 3σ$ above the mean.
• From the Empirical Rule, 99.7% of data lies within 3 standard deviations.
• Half of this percentage (from $-∞$ to $+3σ$) lies below 78 ounces.

Thus, approximately 99.85% of widget weights lie below 78 ounces.

Let me know if you want any clarifications or further details. Here are some related questions you can consider:

1. What percentage of widget weights lie between 48 and 60 ounces?
2. What percentage of widget weights lie above 72 ounces?
3. What range covers 68% of the widget weights?
4. How much of the widget weights are greater than 66 ounces?
5. What is the empirical rule used for in real-life applications?

Tip: The Empirical Rule applies specifically to normal (bell-shaped) distributions and can help you estimate probabilities without complex calculations.