Stats: Standard Deviation and Empirical Rule Essay

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Q 2.20
SOLUTION

The measurement class that represents respondents who mentioned that their company does not outsource any of their computer security functions contains the highest proportion of respondents which was equal to 61%.

6% of the 609 respondents indicated that they outsource between 20% and 40% of their computer security functions.

6% of the 609 respondents outsourced at least 40% of their computer security functions.

88% of the 609 respondents outsourced less that 20% of their computer security functions.

Q 2.30
SOLUTION

Stem-and-Leaf Display: TIME

The following is a stem-and-leaf display for the length of time in bankruptcy for all 49 companies:

Stem-and-leaf of TIME
N = 49
Leaf Unit = 0.10

(26) 1 00001122222344444445555679 23 2 11446799 15 3 002899 9 4 11125 4 5 24 2 6 2 7 8 1 8 1 9 10 1

Summary of information from stem-and-leaf display

We can gather from the stem-and-leaf display that more than half of the 49 companies were bankrupt for less than 2 months. There were 19 companies which spent between 2 and 6 months in bankruptcy and only 2 companies remained bankrupt for more than 6 months.

Histogram of TIME

A Histogram with multiple graphs can be used to show a comparison of the time-in-bankruptcy for the 3 types of prepack firms.

Descriptive Statistics: TIME

Stem-and-leaf of TIME
N = 49 Leaf Unit = 0.10

(26) 1 00001122222344444445555679 23 2 11446799 15 3 002899 9 4 11125 4 5 24 2 6 2 7 8 1 8 1 9 1 10 1

The companies that were reorganized through a leveraged buyout are identified in red on the stem-and-leaf display. It shows that out of the 20 companies that were reorganized through leverage, 13 companies remained bankrupt for less than 2 months.

Q 2.76
SOLUTION

As per the question, we have the mean (x ̅) and standard deviation (s) x ̅=4.25 s = 12.02

We will use Chebyshev’s rule to calculate the interval which will contain at least 75% of the firms. As per Chebyshev’s rule, at least 3⁄4th or 75% of the sample falls between 2 standard deviations of the mean.

The range of the firms within 2 standard deviations of the mean is represented as:

x ̅ ± 2s = (x ̅-2s,x ̅+2s) = (4.25 – 2(12.02), 4.25 + 2(12.02)) = (-19.79, 28.29)

Since the number of blogs cannot be negative, therefore the interval which will most likely contain 75% of the firms will be (0, 28.29).

As per the given data, I expect the distribution of the number of blogs to be skewed right. We know that the mean of the data is 4.25 and that the range for 75% of the data is (0, 28.29). Now, since the number of blogs cannot be negative it means that the majority of the responses will be between 0 and the mean. This will in turn mean that the median will be on the left side of the mean, which implies that the distribution will be right skewed.

Q 2.80
SOLUTION

This is a histogram of 49 companies that filed for bankruptcy: The distribution graph does not seem to be symmetric, so we cannot apply the empirical rule for this distribution.

The numerical descriptive statistics for the data set is shown below:

Descriptive Statistics: TIME

Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum
TIME 49 0 2.549 0.261 1.828 1.000 1.350 1.700 3.500 10.100

Since we cannot apply empirical rule in this scenario, we will use Chebyshev’s rule to construct an interval that captures at least 75% of the bankruptcy time.

As per Chebyshev’s rule, at least 75% of the data is represented by

x ̅ ± 2s = (x ̅-2s,x ̅+2s) = (2.549 – 2(1.828), 2.549 + 2(1.828)) = (-1.107, 6.205)

Since the time for which a firm has been bankrupt cannot be negative, therefore the interval which will most likely contain 75% of the
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