91影视

A survey was conducted by the management accounting to the members of the Institute of Management Accountants. Here the number of members n=2112. The median of salary distribution is 50000. The 20^th percentile of the salary is 35100. The 80^th percentile of the salary distribution is 73000.

a.

Let鈥檚 consider by the Chebyshev鈥檚 theorem, at least$\left(1-\frac{1}{{\mathrm{k}}^2}\right)$ of the observations fall within the k standard deviations of the mean.

So,

$\begin{array}{rcl}\left(1-\frac{1}{{\mathrm{k}}^2}\right)& =& 0.60\\ & 鈬�& \frac{1}{{\mathrm{k}}^2}=1-0.60\\ & 鈬�& {\mathrm{k}}^2=\frac{1}{0.40}\\ & 鈬�& \mathrm{k}=\sqrt{2.5}\\ & 鈬�& k=1.58\end{array}$

Therefore,

$\begin{array}{rcl}\mathrm{s}& =& \frac{{80}^{\mathrm{th}}\mathrm{percentile}-{20}^{\mathrm{th}}\mathrm{percentile}}{2\mathrm{k}}\\ & =& \frac{73000-35100}{2脳1.58}\\ & =& 11985.32\end{array}$

Thus, the sample standard deviation is 11985.32.

Now for a confidence coefficient,

$\begin{array}{rcl}\left(1-\mathrm{伪}\right)& =& 0.98\\ & 鈬�& \mathrm{伪}=1-0.98\\ & 鈬�& \mathrm{伪}=0.02\\ & 鈬�& \frac{\mathrm{伪}}{2}=0.01\end{array}$

Therefore, the z-statistics is

$\begin{array}{rcl}{\mathrm{Z}}_{1-\frac{\mathrm{伪}}{2}}& =& {\mathrm{Z}}_{1-0.01}\\ & =& {\mathrm{Z}}_{0.99}\\ & =& 2.33\end{array}$

So, the required minimum sample size is,

$\begin{array}{rcl}\mathrm{n}& =& \left[\frac{{\mathrm{Z}}_{0.99}{\left(\mathrm{s}\right)}^2}{\mathrm{ME}}\right]\\ \mathrm{n}& =& \left[\frac{2.33脳{\left(11985.32\right)}^2}{2000}\right]\\ & 鈮�& 195\end{array}$

Thus, the required sample size is approximately 195.

b.

Referring to the first part of part a.

At first, let鈥檚 consider there at least$\left(1-\frac{1}{{\mathrm{k}}^2}\right)$ of the observations fall within k standard deviations of the mean by Chebyshev鈥檚 inequality. Then calculate the value of k and then calculate the sample standard deviation by the formula,

$\mathrm{s}=\frac{{80}^{\mathrm{th}}\mathrm{percentile}-{20}^{\mathrm{th}}\mathrm{percentile}}{2\mathrm{k}}$.

c.

There is considered only one assumption that is the estimate of the standard deviation is accurate.