Questions: Using the empirical rule to identify values and percentages of a normal... 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. 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.

Using the empirical rule to identify values and percentages of a normal...

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. 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: Using the empirical rule to identify values and percentages of a normal... 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. 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: Find the value of V

V is located at the highest point of the curve, which corresponds to the mean of the distribution. The problem states that the mean is 189.7 min. Therefore, V = 189.7.

Step 2: Find the values of U and W

U and W are equidistant from V. The standard deviation is 20.2. The shaded area represents the area within one standard deviation of the mean. Therefore: U = V - standard deviation = 189.7 - 20.2 = 169.5 W = V + standard deviation = 189.7 + 20.2 = 209.9

Step 3: Find the percentage of the total area shaded

According to the empirical rule (also known as the 68-95-99.7 rule), approximately 68% of the data in a normal distribution falls within one standard deviation of the mean. The shaded region represents the area within one standard deviation of the mean (from U to W). Therefore, the shaded area represents approximately 68% of the total area.