91Ó°ÊÓ

Find the coefficient of variation for each of the two samples; then compare the variation. (The same data were used in Section 3-1.) Theft Listed below are amounts (in millions of dollars) collected from parking meters by Brinks and others in New York City during similar time periods. A larger data set was used to convict five Brinks employees of grand larceny. The data were provided by the attorney for New York City, and they are listed on the DASL Website. Do the two samples appear to have different amounts of variation? \(\begin{array}{lllllllllll} \text { Collection Contractor Was Brinks } & 1.3 & 1.5 & 1.3 & 1.5 & 1.4 & 1.7 & 1.8 & 1.7 & 1.7 & 1.6 \\ \text { Collection Contractor Was Not Brinks } & 2.2 & 1.9 & 1.5 & 1.6 & 1.5 & 1.7 & 1.9 & 1.6 & 1.6 & 1.8 \end{array}\)

Step by step solution

01

- Calculate the mean for each sample

The mean for Brinks: \(\bar{x}_1 = \frac{1.3 + 1.5 + 1.3 + 1.5 + 1.4 + 1.7 + 1.8 + 1.7 + 1.7 + 1.6}{10}\) The mean for non-Brinks: \(\bar{x}_2 = \frac{2.2 + 1.9 + 1.5 + 1.6 + 1.5 + 1.7 + 1.9 + 1.6 + 1.6 + 1.8}{10}\)

02

- Calculate the standard deviation for each sample

Standard deviation for Brinks: \(s_1 = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x}_1)^2}{n-1}}\) Standard deviation for non-Brinks:\(s_2 = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x}_2)^2}{n-1}}\)

03

- Calculate the coefficient of variation (CV) for each sample

Coefficient of variation is given by \(CV = \frac{s}{\bar{x}}\times100\)For Brinks: \(CV_1 = \frac{s_1}{\bar{x}_1}\times100\)For non-Brinks: \(CV_2 = \frac{s_2}{\bar{x}_2}\times100\)

04

- Compare the coefficients of variation

Compare \(CV_1\) and \(CV_2\) to determine which sample has more variability.

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Key Concepts

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

Statistics

Statistics is a branch of mathematics that helps us understand and analyze data. For instance, it allows us to make sense of the collections from parking meters in the exercise. Key statistical measures include the mean and standard deviation. These help us summarize and describe data sets.

In this exercise, we calculate and compare the mean and standard deviation of two samples collected by different contractors. By doing this, we understand the spread and average of the collected amounts. This process assists in verifying if there's a noticeable difference in the variation of data between the two contractors.

Sample Variability

Sample variability refers to the extent to which data values in a sample differ from one another. It’s an important concept in statistics, as it gives insight into the consistency of the data within a sample.

We measure variability using the standard deviation. A smaller standard deviation indicates that data points are close to the mean, showing less variability. Conversely, a larger standard deviation means the data points are spread out over a wider range, indicating higher variability.

For example, if we see high variability in the money collected from parking meters by Brinks compared to non-Brinks contractors, it raises questions about the consistency and reliability of their collection methods.

Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion in a set of values. It tells us how spread out the data points are from the mean. Calculating it involves the following steps:

• Find the mean (average) of the data set.
• Subtract the mean from each data point and square the result.
• Find the average of these squared differences.
• Take the square root of this average.

In this exercise, we calculate the standard deviation for the collections of Brinks and non-Brinks contractors separately. This gives us a clear understanding of the spread of each collection sample, allowing us to compare their consistency.

Mean Calculation

The mean, often referred to as the average, is a measure that provides a central value for a set of numbers. It is calculated by summing up all the data values and then dividing by the number of values. Here’s how you compute the mean:

• Add all the numbers in the data set.
• Divide the sum by the number of data points.

In our example, we've calculated the mean for both Brinks and non-Brinks collections. These means give us a reference point to understand the general level of collected amounts. This average value plays a crucial role in further calculations, like the standard deviation and the coefficient of variation.