91影视

Arranging the data in ascending order,

First, calculating the information needed to construct a box plot for no prefilling vote,

$$\begin{gathered} \text{Q1}\left(\text{None}\right)\text{=}\frac{\text{N+1}}{\text{4}} \\ \text{=}\frac{\text{11+1}}{\text{4}} \\ \text{=}\frac{\text{12}}{\text{4}} \\ ={\text{3}}^{\text{rd}}\text{term=2.6} \end{gathered}$$

$\begin{array}{c}\text{Q2}\left(\text{None}\right)\text{=}\frac{\text{N+1}}{\text{2}}\\ \text{=}\frac{\text{11+1}}{\text{2}}\\ \text{=}\frac{\text{12}}{\text{2}}\\ {\text{=6}}^{\text{th}}\text{term=3.2}\end{array}$

$\begin{array}{c}\text{Q3}\left(\text{None}\right)\text{=}\frac{\text{3}\left(\text{N+1}\right)}{\text{4}}\\ \text{=}\frac{\text{3}\left(\text{11+1}\right)}{\text{4}}\\ \text{=}\frac{\text{36}}{\text{4}}\\ {\text{=9}}^{\text{th}}\text{term=4}.2\end{array}$

$\begin{array}{c}{\text{IQR=Q}}_{\text{U}}{\text{鈥换}}_{\text{L}}\\ \text{=4.2鈥�2.6}\\ \text{=1.6}\end{array}$

$\begin{array}{c}\text{LowerInnerFence}={\text{Q}}_{\text{L}}鈭�\text{1}.\text{5}\left(\text{IQR}\right)\\ =\text{2}.\text{6}鈥�\text{1}.\text{5}(\text{1}.\text{6})\\ =\text{2}.\text{6}鈥�\text{2}.\text{4}\\ =0.\text{2}\end{array}$

$\begin{array}{c}\text{UpperInnerFence}={\text{Q}}_{\text{U}}+\text{1}.\text{5}\left(\text{IQR}\right)\\ =\text{4}.\text{2}+\text{1}.\text{5}(\text{1}.\text{6})\\ =\text{4}.\text{2}+\text{2}.\text{4}\\ =\text{6}.\text{6}\end{array}$

Second, calculating the data needed for a box plot of pre-pack votes

$$\begin{gathered} \text{Q1}\left(\text{prepack}\right)\text{=}\frac{\text{N+1}}{\text{4}} \\ \text{=}\frac{\text{27+1}}{\text{4}} \\ \text{=}\frac{\text{28}}{\text{4}} \\ {\text{=7}}^{\text{th}}\text{term=1.2} \end{gathered}$$

$$\begin{gathered} \text{Q2}\left(\text{prepack}\right)\text{=}\frac{\text{N+1}}{\text{2}} \\ \text{=}\frac{\text{27+1}}{\text{2}} \\ \text{=}\frac{\text{28}}{\text{2}} \\ {\text{=14}}^{\text{th}}\text{term=1.4} \end{gathered}$$

$$\begin{gathered} \text{Q3}\left(\text{prepack}\right)\text{=}\frac{\text{3}\left(\text{N+1}\right)}{\text{4}} \\ \text{=}\frac{\text{3}\left(\text{27+1}\right)}{\text{4}} \\ \text{=}\frac{\text{84}}{\text{4}} \\ {\text{=21}}^{\text{th}}\text{term=2.1} \end{gathered}$$

$\begin{array}{c}{\text{IQR=Q}}_{\text{U}}{\text{鈥换}}_{\text{L}}\\ \text{=2.1鈥�1.2}\\ \text{=0.9}\end{array}$

$\begin{array}{c}\text{LowerInnerFence}={\text{Q}}_{\text{L}}鈭�\text{1}.\text{5}\left(\text{IQR}\right)\\ =\text{1}.\text{2}鈥�\text{1}.\text{5}\left(0.\text{9}\right)\\ =\text{1}.\text{2}鈥�\text{1}.\text{35}\\ =鈭�0.\text{15}\end{array}$

$\begin{array}{c}{\text{UpperInnerFence=Q}}_{\text{U}}\text{+1.5}\left(\text{IQR}\right)\\ \text{=2.1+1.5}\left(\text{0.9}\right)\\ \text{=2.1+1.35}\\ \text{=3.45}\end{array}$

Third, computing the data for joint votes,

$$\begin{gathered} \text{Q1}\left(\text{Joint}\right)\text{=}\frac{\text{N+1}}{\text{4}} \\ \text{=}\frac{\text{11+1}}{\text{4}} \\ \text{=}\frac{\text{12}}{\text{4}} \\ {\text{=3}}^{\text{rd}}\text{term=1.4} \end{gathered}$$

$$\begin{gathered} \text{Q2(Joint)=}\frac{\text{N+1}}{\text{2}} \\ \text{=}\frac{\text{11+1}}{\text{2}} \\ \text{=}\frac{\text{12}}{\text{2}} \\ {\text{=6}}^{\text{th}}\text{term=1.5} \end{gathered}$$

$$\begin{gathered} \text{Q3(Joint)=}\frac{\text{3}\left(\text{N+1}\right)}{\text{4}} \\ \text{=}\frac{\text{3}\left(\text{11+1}\right)}{\text{4}} \\ \text{=}\frac{\text{36}}{\text{4}} \\ {\text{=9}}^{\text{th}}\text{term=4.5} \end{gathered}$$

$\begin{array}{c}\text{IQR}={\text{Q}}_{\text{U}}鈥�{\text{Q}}_{\text{L}}\\ =\text{4}.\text{5}鈥�\text{1}.\text{4}\\ =\text{3}.\text{1}\end{array}$

$\begin{array}{c}\text{LowerInnerFence}={\text{Q}}_{\text{L}}鈭�\text{1}.\text{5}\left(\text{IQR}\right)\\ =\text{1}.\text{4}鈥�\text{1}.\text{5}(\text{3}.\text{1})\\ =\text{1}.\text{4}鈥�\text{4}.\text{65}\\ =鈭�\text{3}.\text{25}\end{array}$

$\begin{array}{c}\text{UpperInnerFence}={\text{Q}}_{\text{U}}+\text{1}.\text{5}\left(\text{IQR}\right)\\ =\text{4}.\text{5}+\text{1}.\text{5}(\text{3}.\text{1})\\ =\text{4}.\text{5}+\text{4}.\text{65}\\ =\text{9}.\text{15}\end{array}$

Finally, constructing the box plot,

$$\begin{gathered} \text{Median}\left(\text{None}\right)\text{=}\frac{\text{N+1}}{\text{2}} \\ \text{=}\frac{\text{11+1}}{\text{2}} \\ \text{=}\frac{\text{12}}{\text{2}} \\ {\text{=6}}^{\text{th}}\text{term=3.2} \end{gathered}$$

$$\begin{gathered} \text{Median}\left(\text{prepack}\right)\text{=}\frac{\text{N+1}}{\text{2}} \\ \text{=}\frac{\text{27+1}}{\text{2}} \\ \text{=}\frac{\text{28}}{\text{2}} \\ {\text{=14}}^{\text{th}}\text{term=1.4} \end{gathered}$$

$$\begin{gathered} \text{Median}\left(\text{Joint}\right)\text{=}\frac{\text{N+1}}{\text{2}} \\ \text{=}\frac{\text{11+1}}{\text{2}} \\ \text{=}\frac{\text{12}}{\text{2}} \\ {\text{=6}}^{\text{th}}\text{term=1.5} \end{gathered}$$

Bankruptcy time for firms with a pre-pack is lower than the other two types of firms. The IQR is the highest for firms with a joint pack, followed by the firms with no prefiling.

The standard deviation and the IQR for the firms with no prefiling are 2.47 and 1.6, respectively. A very significant gap between the two numbers indicates the impact of outliers on the standard deviation.

The IQR (0.9) and the standard deviation (0.96) for pre-pack firms are almost the same. Whereas the IQR (3.1) and standard deviation (1.72) for the joint group are pretty far from each other, implying outliers' impact.

Based on the boxplots created in step 1, we can say that there are 2 outliers (7.8, 10.1) in the no prefiling group. And 3 outliers (3.8, 4.1, 4.1) in the group of firms with pre-pack.