The question is about the Empirical Rule and using it to understand the distribution of widget weights.

Given:

• Mean ($\mu$) = 61 ounces
• Standard Deviation ($\sigma$) = 5 ounces

The Empirical Rule states that for a normal distribution:

• 68% of data falls within 1 standard deviation of the mean $(\mu \pm \sigma)$
• 95% of data falls within 2 standard deviations $(\mu \pm 2\sigma)$
• 99.7% of data falls within 3 standard deviations $(\mu \pm 3\sigma)$

Now let's solve both parts:

(a) 99.7% of the widget weights lie between:

Using 3 standard deviations:

• Lower bound = $\mu - 3\sigma = 61 - 3(5) = 61 - 15 = 46$ ounces
• Upper bound = $\mu + 3\sigma = 61 + 3(5) = 61 + 15 = 76$ ounces

Answer: 99.7% of the widget weights lie between 46 and 76 ounces.

(b) What percentage of the widget weights lie between 51 and 76 ounces?

1. For 51 ounces, calculate how many standard deviations it is from the mean:

   • $z = \frac{51 - 61}{5} = \frac{-10}{5} = -2$ (This is 2 standard deviations below the mean.)

2. For 76 ounces, we already calculated that it's 3 standard deviations above the mean.

From the Empirical Rule:

• Between $\mu - 2\sigma$ and $\mu + 3\sigma$ (from 51 to 76 ounces), we need to combine the percentages:
  • 95% of the data lies between $\mu - 2\sigma$ and $\mu + 2\sigma$ (i.e., between 51 and 71 ounces).
  • The remaining 2.5% (half of the 5% outside the 95%) lies between 71 and 76 ounces.

Thus, 97.5% of the widget weights lie between 51 and 76 ounces.

Would you like more details on how the Empirical Rule works, or do you have any questions?

Here are some related questions to consider:

1. What is the formula for calculating z-scores and how can they be used to find percentages?
2. How would the results change if the standard deviation increased or decreased?
3. How is the Empirical Rule applied in real-world manufacturing processes?
4. What percentage of weights fall outside 3 standard deviations from the mean?
5. Can the Empirical Rule be applied to distributions that are not normal?

Tip: The Empirical Rule only applies to normal (bell-shaped) distributions; for other shapes, different techniques are needed.