This problem involves a normal distribution with a mean of 189.7 minutes and a standard deviation of 20.2 minutes. You're asked to apply the empirical rule, also known as the 68-95-99.7 rule, to the distribution.

Let's break it down step by step:

1. The Empirical Rule:

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

2. What We Know:

   • The mean $V = 189.7$ minutes.
   • The standard deviation $\sigma = 20.2$ minutes.
   • The values $U$ and $W$ represent points equidistant from $V$.

Calculating $U$, $V$, and $W$:

• $V$ is already given as the mean $189.7$ minutes.
• The points $U$ and $W$ correspond to some number of standard deviations from the mean.

Using the Empirical Rule:

• If $U$ and $W$ are 2 standard deviations away from $V$, they will represent the interval containing 95% of the data.
• To calculate the range: $U = V - 2\sigma = 189.7 - 2(20.2) = 149.3 \, \text{minutes}$ $W = V + 2\sigma = 189.7 + 2(20.2) = 230.1 \, \text{minutes}$

This means that the shaded area likely corresponds to 95% of the total area under the curve.

Conclusion:

• $U = 149.3$ minutes
• $V = 189.7$ minutes (mean)
• $W = 230.1$ minutes
• The shaded area corresponds to 95% of the total area under the normal curve.

Let me know if you would like further clarifications, or if you have questions! Here are 5 related questions you could explore:

1. How is the empirical rule applied in different fields like economics or biology?
2. What happens to the shape of the normal distribution when the standard deviation changes?
3. How do you calculate probabilities for ranges not exactly 1, 2, or 3 standard deviations from the mean?
4. Why is the area under the curve of a normal distribution equal to 1?
5. How does the empirical rule relate to the concept of standard scores (z-scores)?

Tip: Always visualize normal distribution problems to better understand the relationships between the mean, standard deviation, and data spread.