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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.

The lengths of movie files that are available for streaming are modeled using the normal distribution shown below. The mean of the distribution is 189.7 min and the standard deviation is 20.2 min. In the figure, V is a number along the axis and is under the highest part of the curve. U and W are numbers along the axis that are each the same distance away from V. Use the empirical rule to choose the best value for the percentage of the area under the curve that is shaded, and find the values of U, V, and W.

Explore the application of the empirical rule to a normal distribution with a mean of 189.7 minutes and a standard deviation of 20.2 minutes. Learn how to find the values of U, V, and W and calculate the shaded area representing 95% of the total data.

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