Let's address each question using the Empirical Rule, which is applicable to a normal distribution:

1. Empirical Rule Overview:
   • 68% of data lies within 1 standard deviation of the mean.
   • 95% of data lies within 2 standard deviations of the mean.
   • 99.7% of data lies within 3 standard deviations of the mean.

Given:

• Mean = 61 ounces
• Standard Deviation = 3 ounces

a) 68% of the widget weights lie between ______ and ______.

68% of the data lies within 1 standard deviation of the mean. This range is calculated as:

• Mean - 1 Standard Deviation = 61 - 3 = 58 ounces
• Mean + 1 Standard Deviation = 61 + 3 = 64 ounces

Thus, 68% of the widget weights lie between 58 and 64 ounces.

b) What percentage of the widget weights lie between 55 and 64 ounces?

• Calculate the number of standard deviations from the mean:
  • For 55 ounces: $\frac{61 - 55}{3} = 2$ standard deviations below the mean.
  • For 64 ounces: $\frac{64 - 61}{3} = 1$ standard deviation above the mean.

The percentage of weights between 2 standard deviations below the mean and 1 standard deviation above the mean can be calculated as:

• 47.5% (for weights between the mean and 2 standard deviations below) + 34% (for weights between the mean and 1 standard deviation above) = 81.5%.

Thus, 81.5% of the widget weights lie between 55 and 64 ounces.

c) What percentage of the widget weights lie below 70?

• Calculate the number of standard deviations from the mean:
  • $\frac{70 - 61}{3} = 3$ standard deviations above the mean.

According to the Empirical Rule, approximately 99.7% of the data lies within 3 standard deviations. Therefore, approximately 99.85% of the data lies below 70 ounces (since the remaining 0.15% is split equally in the tails).

Thus, approximately 99.85% of the widget weights lie below 70 ounces.

Would you like any further details or have additional questions?

Related Questions:

1. What percentage of widget weights lie between 58 and 67 ounces?
2. What percentage of widget weights lie above 64 ounces?
3. What percentage of widget weights lie between 61 and 67 ounces?
4. What is the probability that a widget weighs more than 70 ounces?
5. What is the range of widget weights that includes 95% of the distribution?

Tip:

Always sketch a bell curve when working with normal distributions to visualize the percentages more effectively.