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Ch. 11 - Goodness-of-Fit and Contingency Tables
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoTriola 14th EditionCh. 11 - Goodness-of-Fit and Contingency TablesProblema 11.1.1a
Capitolo 11, Problema 11.1.1a

Cybersecurity The table below lists the frequency of leading digits of Internet traffic interarrival times for a computer, along with the percentages of each leading digit expected with Benford’s law.


a. Identify the general notation used for observed and expected values.


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The general notation for observed values is typically denoted as O_i, where 'i' represents the category or group (in this case, the leading digit). For example, O_1 represents the observed frequency for the leading digit 1.
The general notation for expected values is typically denoted as E_i, where 'i' represents the category or group. For example, E_1 represents the expected frequency for the leading digit 1 based on Benford's Law.
To calculate the expected frequency (E_i) for each leading digit, multiply the total number of observations by the percentage given in Benford's Law for that digit. For example, E_1 = Total Observations × 30.1%.
The total number of observations can be calculated by summing all the observed frequencies (O_i) for the leading digits. For example, Total Observations = 76 + 62 + 29 + 33 + 19 + 27 + 28 + 21 + 22.
Once the expected frequencies (E_i) are calculated, they can be compared to the observed frequencies (O_i) to analyze how closely the data follows Benford's Law.

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Observed Values

Observed values refer to the actual data collected from a study or experiment. In the context of the table, these are the leading digits of interarrival traffic times, which represent the frequency of each digit as it appears in the dataset. Understanding observed values is crucial for comparing them against expected values to identify patterns or anomalies.
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Expected Value (Mean) of Random Variables

Expected Values

Expected values are theoretical frequencies predicted by a statistical model or law, such as Benford's Law in this case. Benford's Law states that in many naturally occurring datasets, smaller leading digits occur more frequently than larger ones. This concept is essential for evaluating how well the observed data aligns with theoretical expectations, which can indicate whether the data follows a specific distribution.
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Expected Value (Mean) of Random Variables

Benford's Law

Benford's Law is a principle that predicts the frequency distribution of leading digits in many real-life sets of numerical data. According to this law, the number 1 appears as the leading digit about 30% of the time, while larger digits appear less frequently. This law is often used in fraud detection and data analysis, making it important for understanding the significance of the observed and expected values in the context of the given question.
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