Let's analyze the problem step by step:

1. Identify the given values:

   • Mean ($\mu$) = 76.5
   • Standard deviation ($\sigma$) = 5.2

2. Understand the Empirical Rule:

   • The Empirical Rule (68-95-99.7 Rule) states that for a normal distribution:
     • 68% of the data falls within 1 standard deviation ($\mu \pm 1\sigma$).
     • 95% of the data falls within 2 standard deviations ($\mu \pm 2\sigma$).
     • 99.7% of the data falls within 3 standard deviations ($\mu \pm 3\sigma$).

3. Identify $V$, $U$, and $W$:

   • $V$ is at the peak of the normal curve, meaning $V = \mu = 76.5$.
   • $U$ and $W$ are equidistant from $V$, meaning they are symmetrically located at one standard deviation away.

4. Calculate $U$ and $W$:

   • $U = \mu - \sigma = 76.5 - 5.2 = 71.3$
   • $W = \mu + \sigma = 76.5 + 5.2 = 81.7$

5. Determine the percentage of area shaded:

   • The shaded region represents the area between one standard deviation on either side of the mean.
   • According to the Empirical Rule, about 68% of the total area under the normal curve is within this range.

Final Answers:

• $U = 71.3$, $V = 76.5$, $W = 81.7$
• Percentage of total area shaded = 68%

Would you like a more detailed explanation or have any questions?

Related Questions:

1. What percentage of the area is covered within two standard deviations from the mean?
2. How would the values of $U$ and $W$ change if the standard deviation were larger?
3. What percentage of data falls outside of one standard deviation from the mean?
4. How do we compute probabilities using the normal distribution formula instead of the empirical rule?
5. How does changing the mean affect the normal curve's position?

Tip:

When solving normal distribution problems, sketching the curve and marking key values (mean, standard deviations) helps visualize the problem better.