Question

random variables and distributions using the empirical rule to identify values and percentages of a normal... the heights of 8th graders are modeled using the normal distribution shown below. the mean of the distribution is 60.6 in and the standard deviation is 0.9 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, and about 99.7% lies within 3 standard deviations.

Step2: Determine the value of \(V\)

Since \(V\) is under the highest part of the normal - curve (the mean of the distribution), and the mean \(\mu = 60.6\) in, so \(V = 60.6\) in.

Step3: Calculate \(U\) and \(W\)

We know that \(U=\mu-\sigma\) and \(W = \mu+\sigma\) (because \(U\) and \(W\) are 1 standard deviation away from \(V\)). Given \(\mu = 60.6\) in and \(\sigma=0.9\) in.
\(U=60.6 - 0.9=59.7\) in
\(W=60.6 + 0.9=61.5\) in

Step4: Determine the percentage of the shaded area

Since the shaded area is within 1 standard deviation of the mean, by the empirical rule, the percentage of the area under the curve that is shaded is 68%.

Answer:

Percentage of total area shaded: 68%
\(U = 59.7\) in, \(V = 60.6\) in, \(W = 61.5\) in