91Ó°ÊÓ

Florida Power \& Light (FP\&L) Company has enjoyed a reputation for quickly fixing its electric system after storms. However, during the hurricane seasons of 2004 and \(2005,\) a new reality was that the company's historical approach to emergency electric system repairs was no longer good enough (The Wall Street Journal, January 16,2006 ). Data showing the days required to restore electric service after seven hurricanes during 2004 and 2005 follow. $$\begin{array}{lc} \text { Hurricane } & \text { Days to Restore Service } \\ \text { Charley } & 13 \\ \text { Frances } & 12 \\ \text { Jeanne } & 8 \\ \text { Dennis } & 3 \\ \text { Katrina } & 8 \\ \text { Rita } & 2 \\ \text { Wilma } & 18\end{array}$$ Based on this sample of seven, compute the following descriptive statistics: a. Mean, median, and mode b. Range and standard deviation c. Should Wilma be considered an outlier in terms of the days required to restore electric service? d. The seven hurricanes resulted in 10 million service interruptions to customers. Do the statistics show that FP\&L should consider updating its approach to emergency electric system repairs? Discuss.

Step by step solution

01

Organize the data

First, let's list the number of days it took to restore power after each hurricane: \( \{13, 12, 8, 3, 8, 2, 18\} \). This list will be used to find the mean, median, mode, range, and standard deviation.

02

Calculate the Mean

The mean is the average of a set of numbers. To obtain this, sum up all the days, then divide by 7 (the number of hurricanes). \[ \text{Mean} = \frac{13 + 12 + 8 + 3 + 8 + 2 + 18}{7} = \frac{64}{7} = 9.14 \] The mean number of days is approximately 9.14 days.

03

Determine the Median

The median is the middle number in an ordered data set. Arrange the data in ascending order: \( \{2, 3, 8, 8, 12, 13, 18\} \). The middle number is the fourth number, which is 8. The median is 8 days.

04

Identify the Mode

The mode is the number that appears most frequently in the dataset. Here, the number 8 appears twice, and no other number repeats. Thus, the mode is 8 days.

05

Calculate the Range

The range is the difference between the maximum and minimum values. In the dataset, the highest is 18 and the lowest is 2. \[ \text{Range} = 18 - 2 = 16 \] The range is 16 days.

06

Calculate the Standard Deviation

The standard deviation measures the amount of variation in a set. First, find the variance.\[ \text{Variance} = \frac{(13 - 9.14)^2 + (12 - 9.14)^2 + (8 - 9.14)^2 + (3 - 9.14)^2 + (8 - 9.14)^2 + (2 - 9.14)^2 + (18 - 9.14)^2}{7} = 24.98 \] Then, take the square root of the variance to find the standard deviation.\[ \text{Standard Deviation} = \sqrt{24.98} \approx 4.99 \]

07

Assess Wilma as an Outlier

To determine if Wilma should be considered an outlier, use the interquartile range (IQR) method. The lower quartile (Q1) is between the 2nd and 3rd terms, so \( Q1 = 3 \). The upper quartile (Q3) is between the 5th and 6th terms, so \( Q3 = 13 \). Calculate the IQR: \( \text{IQR} = 13 - 3 = 10 \). An outlier is typically a value greater than \( Q3 + 1.5 \times \text{IQR} = 13 + 15 = 28 \) or less than \( Q1 - 1.5 \times \text{IQR} = 3 - 15 = -12 \). Since 18 is not greater than 28, Wilma is not considered an outlier.

08

Interpretation of Results

The descriptive statistics show a reasonable spread of data around the mean, with relatively high variability as indicated by the standard deviation. The range of 16 days and the high number of service interruptions suggest that FP&L's current approach may not be sufficient. Updates to repair strategies could potentially provide more efficient restorations and better prepare for extreme events like Wilma.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation

To understand the process of calculating the mean, or average, you sum all the values in your data set and then divide by the number of values. This gives you an idea of the typical value in the dataset. For example, in the hurricane data provided, the days it took to restore service were \{13, 12, 8, 3, 8, 2, 18\}. By summing these numbers (\(13 + 12 + 8 + 3 + 8 + 2 + 18\)) and dividing by 7, we get \( \text{Mean} = \frac{64}{7} \approx 9.14 \). Thus, on average, it took about 9.14 days to restore electrical service. This mean value helps summarize the overall hurricane restoration effort but doesn't show variations in the data.

Standard Deviation

Standard deviation is a statistical tool that measures the dispersion or spread of a dataset relative to its mean. It helps us understand how much the values in the dataset vary from the average value. A higher standard deviation indicates more variability.In this context, we calculated a standard deviation of approximately 4.99 days. This number tells us that the number of days to restore power typically deviates by about 5 days from the average (mean) restoration time of 9.14 days. To calculate this, we had to first determine the variance, which involves subtracting the mean from each data point, squaring the result, and finding the average of those squared differences.The formula for standard deviation (\sigma) from a dataset is \( \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \), where \(x_i\) is each data point, \(\mu\) is the mean, and \(N\) is the number of observations.Comprehending standard deviation helps FP&L recognize the consistency in restoration times and indicates areas where performance could be improved.

Outlier Detection

Detecting outliers in a dataset can be crucial, as they may represent unusual, potentially problematic, or significant data points. One common method to find outliers is using the Interquartile Range (IQR), which is the range between the first (\(Q1\)) and third quartile (\(Q3\)) values.In our hurricane service restoration data, we found the \(Q1\) to be 3 and \(Q3\) to be 13. The IQR is then \(Q3 - Q1 = 10\). An outlier would typically lie outside \(Q3 + 1.5 \, \text{IQR}\) or below \(Q1 - 1.5 \, \text{IQR}\). Given our calculations, a number greater than 28 or less than -12 could be considered an outlier.However, the value of 18 days, associated with Hurricane Wilma, isn't more than 28, so it's not an outlier according to this technique. Outliers are important to notice because they might indicate abnormal conditions that need attention, errors in data collection, or real shifts in the behavior being measured.

Hurricane Impact Analysis

Analyzing the impact of hurricanes on the restoration of power is vital for improving emergency responses. The data from FP&L highlights varying recovery times following seven hurricanes. This impact analysis can help shape future strategies. FP&L faced significant disruptions, with a total of 10 million service interruptions during these seven hurricanes. The analysis indicates that, on average, restorations took about 9 days, but with significant variability—from 2 to 18 days. Such disparities suggest the company's existing strategies need evaluation and potential improvement. Through statistics, the company can better grasp the frequency and severity of service interruptions, helping them to:

• Enhance resource allocation and preparedness.
• Focus on cycles of improvement in response times.
• Deploy better infrastructure for resilience against major storms.

The insights derived from impact analyses like these are essential for FP&L to update and optimize their emergency repair plans, ensuring quicker response times in future hurricane seasons.