Let's break this down using the Empirical Rule, which applies to a normal distribution:

The mean is 46 ounces, and the standard deviation (σ) is 10 ounces.

a) 68% of the widget weights lie between:

The Empirical Rule states that 68% of the data in a normal distribution lies within 1 standard deviation of the mean.

Thus, the range will be: $46 \, (\text{mean}) \pm 10 \, (\text{standard deviation})$

So, the weights lie between: $46 - 10 = 36 \quad \text{and} \quad 46 + 10 = 56$

Answer: 36 ounces and 56 ounces.

b) What percentage of the widget weights lie between 26 and 56 ounces?

We need to determine how many standard deviations 26 and 56 ounces are from the mean of 46 ounces.

• For 26 ounces: $\frac{46 - 26}{10} = 2 \, (\text{2 standard deviations below the mean})$

• For 56 ounces: $\frac{56 - 46}{10} = 1 \, (\text{1 standard deviation above the mean})$

From the Empirical Rule:

• 95% of data falls within 2 standard deviations of the mean.
• 68% of data falls within 1 standard deviation of the mean.

Since we are looking from -2σ (26 ounces) to +1σ (56 ounces):

• From -2σ to +2σ: 95%
• Half of this is from -2σ to the mean: 47.5%
• From the mean to +1σ: 34%

Thus, the total percentage: $47.5\% + 34\% = 81.5\%$

Answer: 81.5%.

c) What percentage of the widget weights lie above 16 ounces?

We need to determine how many standard deviations 16 ounces is from the mean: $\frac{46 - 16}{10} = 3 \, (\text{3 standard deviations below the mean})$

According to the Empirical Rule:

• 99.7% of data lies within 3 standard deviations (±3σ) of the mean.
• Half of this is 49.85%, representing the data above 16 ounces (as 16 is -3σ).

Thus, the percentage above 16 ounces is approximately 49.85%.

Do you need any more clarifications or further details on these calculations?

Follow-up Questions:

1. What does the Empirical Rule say about 95% of the data distribution?
2. How can we apply the Empirical Rule to data more than 3 standard deviations from the mean?
3. What would be the range for 99.7% of the data?
4. How does standard deviation affect the spread of data in a normal distribution?
5. How can the percentages between ranges be computed for non-standard deviation intervals?

Tip:

Always visualize the bell curve and Empirical Rule percentages to better understand normal distributions.