Questions: The annual amounts of rainfall in a certain region are modeled using the normal distribution shown below. The mean of the distribution is 30.8 cm and the standard deviation is 3.3 cm. In the figure, V is a number along the axis and is under the highest part of the curve. And, 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.

The annual amounts of rainfall in a certain region are modeled using the normal distribution shown below. The mean of the distribution is 30.8 cm and the standard deviation is 3.3 cm. In the figure, V is a number along the axis and is under the highest part of the curve. And, 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.

Transcript text: The annual amounts of rainfall in a certain region are modeled using the normal distribution shown below. The mean of the distribution is 30.8 cm and the standard deviation is 3.3 cm . In the figure, $V$ is a number along the axis and is under the highest part of the curve. And, $U$ 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 $W$.

Solution

Solution Steps

Step 1: Identify the mean and standard deviation

The problem states that the mean of the distribution is 30.8 cm and the standard deviation is 3.3 cm. Since V is located at the highest point of the curve, it corresponds to the mean.

Step 2: Determine the values of U and W using the empirical rule

The empirical rule (or 68-95-99.7 rule) states that for a normal distribution:

Approximately 68% of the data falls within one standard deviation of the mean.
Approximately 95% of the data falls within two standard deviations of the mean.
Approximately 99.7% of the data falls within three standard deviations of the mean.

The shaded region appears to cover the range within two standard deviations of the mean. U is two standard deviations below the mean and W is two standard deviations above the mean. Therefore: $U = V - 2 \times \text{standard deviation} = 30.8 - 2 \times 3.3 = 30.8 - 6.6 = 24.2$ $W = V + 2 \times \text{standard deviation} = 30.8 + 2 \times 3.3 = 30.8 + 6.6 = 37.4$

Step 3: Calculate the percentage of the area shaded

Since the shaded area is within two standard deviations of the mean, based on the empirical rule, approximately 95% of the area under the curve is shaded.