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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a. Enter the observed frequencies in the table above.
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Ch. 11 - Goodness-of-Fit and Contingency Tables
Triola14th EditionElementary StatisticsISBN: 9780137366446
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Table of contents
- Ch. 1 - Introduction to Statistics88
- Ch. 2 - Exploring Data with Tables and Graphs70
- Ch. 3 - Describing, Exploring, and Comparing Data66
- Ch. 4 - Probability112
- Ch. 5 - Discrete Probability Distributions109
- Ch. 6 - Normal Probability Distributions122
- Ch. 7 - Estimating Parameters and Determining Sample Sizes107
- Ch. 8 - Hypothesis Testing75
- Ch. 9 - Inferences from Two Samples94
- Ch. 10 - Correlation and Regression80
- Ch. 11 - Goodness-of-Fit and Contingency Tables29
- Ch. 12 - Analysis of Variance42
Chapter 11, Problem 11.1.1b
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.
b. Identify the observed and expected values for the leading digit of 2.
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with identifying the observed and expected values for the leading digit of 2. The observed values are given in the 'Leading Digits of Interarrival Traffic Times' row, and the expected values are based on Benford's Law percentages.
Step 2: Locate the observed value for the leading digit of 2. From the table, under the column labeled '2' in the 'Leading Digits of Interarrival Traffic Times' row, the observed value is 62.
Step 3: Locate the expected percentage for the leading digit of 2. From the table, under the column labeled '2' in the 'Benford’s Law' row, the expected percentage is 17.6%.
Step 4: Calculate the expected value. Multiply the total number of observations (sum of all observed values) by the expected percentage (17.6%) to find the expected value. Use the formula: Expected Value = Total Observations × (Expected Percentage / 100).
Step 5: Summarize the results. The observed value for the leading digit of 2 is 62, and the expected value can be calculated using the formula in Step 4.
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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.
Observed Values
Observed values refer to the actual counts or frequencies of data points collected from a specific dataset. In the context of the question, the observed value for the leading digit '2' is 62, which represents how many times '2' appears as the leading digit in the interarrival traffic times. These values are essential for comparison against expected values to assess conformity to theoretical distributions.
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Expected Value (Mean) of Random Variables
Expected Values
Expected values are the theoretical frequencies predicted by a statistical model, such as Benford's Law. For the leading digit '2', the expected value is calculated as 17.6% of the total number of observations. This comparison between expected and observed values helps in determining how well the data aligns with the predictions of Benford's Law, which can indicate anomalies or irregularities in the dataset.
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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
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
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?
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.
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.
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