The table below lists leading digits of 317 inter-arrival...

The table below lists leading digits of 317 inter-arrival Internet traffic times for a computer, along with the frequencies of leading digits expected with Benford's law (from Table $11-1$ in the Chapter Problem). a. Identify the notation used for observed and expected values. b. Identify the observed and expected values for the leading digit of $2 .$ c. Use the results from part (b) to find the contribution to the $\chi^{2}$ test statistic from the category representing the leading digit of 2 . $$ \begin{array}{l|c|c|c|c|c|c|c|c|c|} \hline \text { Leading Digit } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \text { Benford's Law } & 30.1 \% & 17.6 \% & 12.5 \% & 9.7 \% & 7.9 \% & 6.7 \% & 5.8 \% & 5.1 \% & 4.6 \% \\ \hline \begin{array}{l} \text { Leading Digits of } \\ \text { Inter-Arrival Traffic Times } \end{array} & 76 & 62 & 29 & 33 & 19 & 27 & 28 & 21 & 22 \\ \hline \end{array} $$

The table below lists leading digits of 317 inter-arrival Internet traffic times for a computer, along with the frequencies of leading digits expected with Benford's law (from Table $11-1$ in the Chapter Problem).
a. Identify the notation used for observed and expected values.
b. Identify the observed and expected values for the leading digit of $2 .$
c. Use the results from part (b) to find the contribution to the $\chi^{2}$ test statistic from the category representing the leading digit of 2 .
$$
\begin{array}{l|c|c|c|c|c|c|c|c|c|}
\hline \text { Leading Digit } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline \text { Benford's Law } & 30.1 \% & 17.6 \% & 12.5 \% & 9.7 \% & 7.9 \% & 6.7 \% & 5.8 \% & 5.1 \% & 4.6 \% \\
\hline \begin{array}{l}
\text { Leading Digits of } \\
\text { Inter-Arrival Traffic Times }
\end{array} & 76 & 62 & 29 & 33 & 19 & 27 & 28 & 21 & 22 \\
\hline
\end{array}
$$

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Key Concepts

Observed Values

These are the actual counts or frequencies recorded from data. In statistical notation, observed counts are often represented by O and denote what has been measured or collected in each category within a study.

Expected Values

Expected values refer to the counts that one would anticipate under a theoretical model or hypothesis. They are typically denoted as E and are calculated by multiplying the model's predicted proportion by the total sample size.

Expected Frequency Calculation

This concept involves converting a theoretical probability or percentage into an expected count by multiplying it by the total number of observations. This step is critical in practical applications because it allows for a direct comparison between observed data and model predictions.

Chi-Squared Test Statistic

The chi-squared test statistic is used in goodness-of-fit tests to measure the discrepancy between observed and expected values. Its computation involves summing the squared differences between observed (O) and expected (E) counts divided by the E values for each category, which helps assess whether the deviations from the model predictions are statistically significant.

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