Questions: The heights of 8th graders are modeled using the normal distribution shown below. The mean of the distribution is 63.6 in and the standard deviation is 1.6 in. 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 heights of 8th graders are modeled using the normal distribution shown below. The mean of the distribution is 63.6 in and the standard deviation is 1.6 in. 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 heights of 8th graders are modeled using the normal distribution shown below. The mean of the distribution is 63.6 in and the standard deviation is 1.6 in . 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: Determine V

V is located at the highest point of the curve, which corresponds to the mean of the distribution. The problem states the mean is 63.6 inches. Therefore, V = 63.6.

Step 2: Determine U and W

The empirical rule (or 68-95-99.7 rule) states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. The shaded region corresponds to the area within one standard deviation of the mean.

The problem states that U and W are equidistant from V. Since the shaded region represents one standard deviation from the mean, we can calculate U and W by subtracting and adding the standard deviation (1.6 inches) from the mean (63.6 inches).

U = 63.6 - 1.6 = 62 W = 63.6 + 1.6 = 65.2

Step 3: Determine the shaded area percentage.

The shaded region represents the area within one standard deviation of the mean. According to the empirical rule, this area corresponds to approximately 68% of the total area under the curve.