Ch. 11 - Goodness-of-Fit and Contingency Tables

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 11 - Goodness-of-Fit and Contingency Tables

Problem 11.1.1c

Chapter 11, Problem 11.1.1c

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.

c. Use the results from part (b) to find the contribution to the x2 test statistic from the category representing the leading digit of 2.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with calculating the contribution to the chi-squared (χ²) test statistic for the category representing the leading digit of 2. The χ² test statistic measures the difference between observed and expected frequencies.

Step 2: Identify the observed frequency (O) for the leading digit of 2. From the table, the observed frequency is 62.

Step 3: Calculate the expected frequency (E) for the leading digit of 2. Use the percentage from Benford's Law (17.6%) and multiply it by the total number of observations. To find the total number of observations, sum all the observed frequencies in the table.

Step 4: Use the formula for the contribution to the χ² test statistic for the leading digit of 2: χ²_contribution = ((O - E)^2) / E. Substitute the observed frequency (O) and the expected frequency (E) into the formula.

Step 5: Simplify the expression to find the contribution to the χ² test statistic for the leading digit of 2. This value will be part of the overall χ² test statistic.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Benford's Law

Benford's Law predicts the frequency distribution of leading digits in many real-life sets of numerical data. According to this law, smaller digits occur as the leading digit more frequently than larger digits. For example, the digit '1' appears as the leading digit about 30.1% of the time, while '9' appears only about 4.6% of the time. This phenomenon is often used in fraud detection and data analysis.

Chi-Squared Test

The Chi-Squared test is a statistical method used to determine if there is a significant difference between the expected and observed frequencies in categorical data. In this context, it helps assess how well the observed leading digits of interarrival traffic times fit the expected frequencies according to Benford's Law. The test statistic is calculated by summing the squared differences between observed and expected values, divided by the expected values.

Contribution to Chi-Squared Statistic

The contribution to the Chi-Squared statistic for a specific category is calculated by taking the squared difference between the observed frequency and the expected frequency for that category, divided by the expected frequency. This contribution helps identify which categories deviate most from what is expected under Benford's Law, providing insight into the data's distribution and potential anomalies.

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Step 2: Calculate Test Statistic

Related Practice

Textbook Question

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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Textbook Question

b. Identify the observed and expected values for the leading digit of 2.

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Textbook Question

Testing Goodness-of-Fit with a Normal Distribution Refer to Data Set 1 “Body Data” in Appendix B for the heights of females.

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c. Using the probabilities found in part (b), find the expected frequency for each category.

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Textbook Question

Testing Goodness-of-Fit with a Normal Distribution Refer to Data Set 1 “Body Data” in Appendix B for the heights of females.

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b. Assuming a normal distribution with mean and standard deviation given by the sample mean and standard deviation, use the methods of Chapter 6 to find the probability of a randomly selected height belonging to each class.

Textbook Question

Questions 6–10 refer to the sample data in the following table, which describes the fate of the passengers and crew aboard the Titanic when it sank on April 15, 1912. Assume that the data are a sample from a large population and we want to use a 0.05 significance level to test the claim that surviving is independent of whether the person is a man, woman, boy, or girl.

Is the hypothesis test left-tailed, right-tailed, or two-tailed?

Textbook Question

Exercises 1–5 refer to the sample data in the following table, which summarizes the frequencies of 500 digits randomly generated by Statdisk. Assume that we want to use a 0.05 significance level to test the claim that Statdisk generates the digits in a way that they are equally likely.

What are the null and alternative hypotheses corresponding to the stated claim?