In this problem, we’re asked to use the empirical rule (also known as the 68-95-99.7 rule) for a normal distribution, with a mean of 72.4 and a standard deviation of 5.5, to find the values of $U$, $V$, and $W$ and determine the percentage of the shaded area under the curve.

Step-by-Step Solution:

1. Identify the Positions of $U$, $V$, and $W$:

   • The empirical rule tells us that:
     • Approximately 68% of the data falls within one standard deviation from the mean.
     • Approximately 95% of the data falls within two standard deviations from the mean.
     • Approximately 99.7% of the data falls within three standard deviations from the mean.

2. Calculate $U$, $V$, and $W$:

   • Since $V$ is located at the mean, $V = 72.4$.
   • $U$ and $W$ are symmetric around the mean, each representing one standard deviation away from the mean.
     • $U = 72.4 - 5.5 = 66.9$
     • $W = 72.4 + 5.5 = 77.9$

3. Determine the Shaded Area Percentage:

   • Since $U$ and $W$ are one standard deviation away from the mean, the shaded area represents approximately 68% of the total area under the curve, according to the empirical rule.

Answer Summary:

• $U = 66.9$
• $V = 72.4$
• $W = 77.9$
• Shaded area percentage: 68%

Would you like a further breakdown of these calculations or additional information on the empirical rule?

Here are five related questions to explore this topic further:

1. How would the values change if the standard deviation were different?
2. What would the values of $U$ and $W$ be for two standard deviations from the mean?
3. How does the empirical rule help in understanding normal distributions in real-world data?
4. What percentage of the area lies within three standard deviations of the mean?
5. How can we use the empirical rule to approximate probabilities for different ranges in a normal distribution?

Tip: The empirical rule is a quick way to estimate probabilities and can be useful in understanding the spread of data in a normal distribution.