Question

the heights of 8th - graders are modeled using the normal distribution shown below. the mean of the distribution is 63.1 in and the standard deviation is 1.5 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. percentage of total area shaded: select

Explanation:

Step1: Recall the empirical rule

The empirical rule for a normal - distribution states that about 68% of the data lies within 1 standard deviation of the mean, about 95% lies within 2 standard deviations of the mean, and about 99.7% lies within 3 standard deviations of the mean.

Step2: Identify the mean and standard deviation

Given that the mean $\mu = 63.1$ inches and the standard deviation $\sigma=1.5$ inches. The value $V$ is the mean, so $V = 63.1$ inches.

Step3: Calculate $U$ and $W$

Since $U$ and $W$ are the same distance from $V$, and if we assume the shaded area represents the middle 95% of the data (2 standard - deviations from the mean), then $U=\mu - 2\sigma$ and $W=\mu + 2\sigma$.
$U=63.1-2\times1.5=63.1 - 3=60.1$ inches.
$W=63.1 + 2\times1.5=63.1+3 = 66.1$ inches.
The percentage of the total area shaded, according to the empirical rule for data within 2 standard deviations of the mean, is 95%.

Answer:

Percentage of total area shaded: 95%
$U = 60.1$ inches
$V = 63.1$ inches
$W = 66.1$ inches