The problem you uploaded involves a normal distribution with a mean of 170.5 minutes and a standard deviation of 19.9 minutes. You're tasked with identifying points $U$, $V$, and $W$, which are spaced equally around the center of the normal distribution (at $V$, the mean), and calculating the percentage of the total shaded area under the curve.

Step-by-step breakdown:

1. Normal Distribution Basics:

   • Mean $\mu = 170.5$ minutes.
   • Standard deviation $\sigma = 19.9$ minutes.

2. Identifying $V$:

   • $V$ is the mean, located under the highest point of the bell curve. So, $V = 170.5$.

3. Identifying $U$ and $W$:

   • $U$ and $W$ are symmetric points around the mean, such that the distance from $V$ to $U$ is the same as from $V$ to $W$. These distances likely correspond to intervals of the standard deviation (based on the "empirical rule").

4. Using the Empirical Rule:

   • 68% of the data falls within one standard deviation ($V - \sigma$ to $V + \sigma$).
   • 95% of the data falls within two standard deviations ($V - 2\sigma$ to $V + 2\sigma$).
   • 99.7% of the data falls within three standard deviations ($V - 3\sigma$ to $V + 3\sigma$).

From the image, it appears the points $U$ and $W$ mark an interval that might correspond to about 95% of the total area under the curve. This suggests that $U = V - 2\sigma$ and $W = V + 2\sigma$.

5. Calculating $U$ and $W$: $U = V - 2\sigma = 170.5 - 2(19.9) = 170.5 - 39.8 = 130.7$ $W = V + 2\sigma = 170.5 + 2(19.9) = 170.5 + 39.8 = 210.3$

Thus, $U = 130.7$, $V = 170.5$, and $W = 210.3$.

6. Percentage of Area Shaded:
   • Since $U$ and $W$ correspond to two standard deviations from the mean, the percentage of the shaded area under the curve is approximately 95%.

Would you like more details on any part of the explanation?

Related Questions:

1. How would the shaded area change if we considered only one standard deviation from the mean?
2. What percentage of data lies outside the range of two standard deviations?
3. If the standard deviation were increased, how would $U$ and $W$ change?
4. How would the shape of the curve change with a smaller standard deviation?
5. Can you explain how the empirical rule applies to non-symmetric distributions?

Tip:

When dealing with normal distributions, memorizing the empirical rule (68-95-99.7) helps in quickly estimating probabilities based on standard deviations.