Let's tackle these questions step by step using the Empirical Rule (also known as the 68-95-99.7 Rule).

Question 9:

We are given the following information:

• Mean (μ) = 37 ounces
• Standard deviation (σ) = 9 ounces
• The distribution is normal (bell-shaped).

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

According to the Empirical Rule, 68% of the data lies within 1 standard deviation of the mean. So, the range is: $\text{Mean} - 1\sigma \text{ to } \text{Mean} + 1\sigma$ $37 - 9 \text{ to } 37 + 9 \Rightarrow 28 \text{ to } 46 \text{ ounces}$ Answer: 28 and 46 ounces

b) What percentage of widget weights lie between 19 and 46 ounces?

Now, we calculate how many standard deviations the values 19 and 46 are from the mean: $\frac{19 - 37}{9} = -2 \quad (\text{2 standard deviations below the mean})$ $\frac{46 - 37}{9} = 1 \quad (\text{1 standard deviation above the mean})$ According to the Empirical Rule:

• From 2 standard deviations below the mean to 1 standard deviation above the mean includes 81.5% of the data. $P(19 \leq X \leq 46) = 81.5\%$ Answer: 81.5%

c) What percentage of the widget weights lie below 64?

Calculate how many standard deviations 64 is from the mean: $\frac{64 - 37}{9} = 3 \quad (\text{3 standard deviations above the mean})$ According to the Empirical Rule:

• 99.7% of the data lies within 3 standard deviations of the mean.
• Half of this (below the mean) is 49.85%, and everything above that is the remaining part: $P(X \leq 64) = 99.85\%$ Answer: 99.85%

Question 10:

We are given the following:

• Mean (μ) = 6.7 inches
• Standard deviation (σ) = 0.8 inches
• We want the range that covers 99.7% of the data, which is 3 standard deviations from the mean.

$\text{Mean} - 3\sigma \text{ to } \text{Mean} + 3\sigma$ $6.7 - 3(0.8) \text{ to } 6.7 + 3(0.8) \Rightarrow 4.3 \text{ to } 9.1 \text{ inches}$ Answer: 4.3 and 9.1 inches

Would you like more details on any of the steps, or do you have any further questions?

Follow-up Questions:

1. How does the Empirical Rule help us approximate probabilities in a normal distribution?
2. What is the meaning of "standard deviation" in the context of this problem?
3. How would the percentages change if the distribution were skewed instead of normal?
4. Can we use the Empirical Rule for distributions that aren't bell-shaped?
5. What other rules or approximations can we use for understanding distributions?

Tip:

Always sketch a quick normal distribution curve when using the Empirical Rule to visualize the percentages within 1, 2, and 3 standard