Chapter 11.1, Problem 31E

To determine

To Describe: the $84$ th percentile in terms of the mean and standard deviation.

Answer to Problem 31E

  $84$ th percentile corresponds to a value about one standard deviation above the mean.

Explanation of Solution

Calculation:

  $84$ th percentile corresponds to a value about one standard deviation above the mean that is $\mu +\sigma$ .That is area shaded under the normal curve as shown.

  

To find area under normal curve, these properties of a normal distribution is used,

• The total area under the normal curve is $1$ .
• About $68\%$ of the area lies within $1$ standard deviation of mean.
• About $95\%$ of the area lies within $2$ standard deviation of mean.
• About $99.7\%$ of the area lies within $3$ standard deviation of mean.

  

Now, comparing the first and second diagrams, the percent of the shaded area is given by shaded area

  $\begin{array}{l}=0.15\%+2.35\%+13.5\%+34\%+34\%\\ =84\%\end{array}$

Hence, $84$ th percentile corresponds to a value about one standard deviation above the mean.

Conclusion:

Hence, the $84$ th percentile corresponds to a value about one standard deviation above the mean.