Salary offers to MBAs.Consider the top salary offer (in thousands of dollars) received by each member of a sample of 50 MBA students who graduated from the Graduate School of Management at Rutgers, the state university of New Jersey. Descriptive statistics and a box plot for the data are shown on the XLSTAT printouts at the top of the next column. [Note:The “+” on the box plot represents the location of the mean.]

a.Find and interpret the z-score associated with the highest salary offer, the lowest salary offer, and the mean salary offer. Would you consider the highest offer to be unusually high? Why or why not?

b.Based on the box plot for this data set, which salary offers (if any) are suspect or highly suspect outliers?

$\begin{array}{rcl}zscorehighestsalary& =& \frac{Max-Mean}{S\tan dardDeviation}\\ & =& \frac{\$75,000-\$52,334}{\$9,224}\\ & =& 2.45\end{array}$

The z-score for the highest salary is 2.45. Because the z-score is less than 3, it is a possible outlier.

$\begin{array}{rcl}zscorelowestsalary& =& \frac{Min-Mean}{S\tan dardDeviation}\\ & =& \frac{\$35,000-\$52,334}{\$9,224}\\ & =& -1.87\end{array}$

The z-score of the lowest salary indicates that it is not an outlier.

$\begin{array}{rcl}zscoremeansalary& =& \frac{Mean-Mean}{S\tan dardDeviation}\\ & =& \frac{\$52,334-\$52,334}{\$9,224}\\ & =& \frac{\$0}{\$9,224}\\ & =& 0\\ & & \end{array}$

The mean salary offer lies within the range and is not an outlier.

No, I would not consider the highest salary offer as unusually high because it is not an outlier and lies well within the 3 standard deviations from the mean range.

Suspect outliers lie between the inner and outer fences.

The inner fences are the first and the third quartiles. Therefore, Q_L = $45,725 and Q_U = $ 58,900.

The outer fences are approx. $75,000 and $35,000.

$\begin{array}{rcl}SuspectOutliers& =& Q_L-1.5\left(IQR\right)and{Q}_L-1.5\left(IQR\right)\\ & =& 45,725-1.5\left(13,175\right)and45,725-3\left(13,175\right)\\ & =& 45,725-19,762.5and45,725-39,525\\ & =& \$25,962.5and\$6,200\end{array}$

$\begin{array}{rcl}\mathrm{Suspect}\mathrm{Outliers}& =& {\mathrm{Q}}_{\mathrm{U}}+1.5\left(\mathrm{IQR}\right)\mathrm{and}{\mathrm{Q}}_{\mathrm{U}}+1.5\left(\mathrm{IQR}\right)\\ & =& 58,900+1.5\left(13,175\right)\mathrm{and}58,900+3\left(13,175\right)\\ & =& 58,900+19,762.5\mathrm{and}58,900+39,525\\ & =& \$78,662\mathrm{and}\$98,425\end{array}$

Any observations that lie between $25,962.5 and $6,200; and $78,662.5 and $98,425 are suspect outliers.

Highly suspect outliers lie above or below the outer fences.

Therefore, any observation above $75,000 and below $35,000 is a highly suspect outlier.