**Understanding Benford's Law in Business Expenses**Benford's law states that the probability distribution of the first digits of many items (e.g., populations and expenses) is not uniform, but has the probabilities shown in this table.Businesses tend to follow Benford's Law because there are generally more small expenses than large expenses. Let's perform a "Goodness of Fit" Chi-Squared hypothesis test (α = 0.05) to see if these values are consistent with Benford's Law. If they are not consistent, there might be embezzlement.Complete the table. The sum of the observed frequencies is 137.| First Digit (X) | Observed Frequency (Counts) | Benford’s Law P(X) | Expected Frequency (Counts) ||-----------------|------------------------------|---------------------|-----------------------------|| 1 | 39 | .301 | || 2 | 29 | .176 | || 3 | 20 | .125 | || 4 | 12 | .097 | || 5 | 7 | .079 | || 6 | 11 | .067 | || 7 | 7 | .058 | || 8 | 6 | .051 | || 9 | 6 | .046 | |Report all answers accurate to three decimal places.**Steps to Complete the Table:**1. Calculate the Expected Frequency (Counts) for each digit by multiplying the total observed frequencies by the probability given by Benford’s Law. - For digit 1: Expected Frequency = 137 * 0.301 = 41.237 - For digit 2: Expected Frequency = 137 * 0.176 = 24.112 - Continue similarly for all digits.2. Once the Expected Frequencies are filled, you can proceed to compute the Chi-square test-statistic (χ²).**Chi-Square Test Computation:**1. Calculate the Chi-square test-statistic (χ²) using the formula: \[ \chi^2 = \sum \frac{(Observed -

Null hypothesis: The given data values are consistent with Benford's Law.Hypothesis hypothesis: The…

Answered: Benford's law states that the…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question 6.1 Please solve the problem with simple probability rules (no Excel)
A skin care company claims that 90% of its customers have seen a dramatic reduction in fine lines and wrinkles after using the company's products for three weeks. What is the probability that 8 out of 12 customers will see dramatic results? What is the probability that at least 8 customers will see these dramatic results? Does the company's claim appear to hold true if only 8 customers out of 12 experience a dramatic decrease in fine lines and wrinkles?
The rates of on-time flights for commercial jets are continuously tracked by the U.S. Department of Transportation. Recently, Southwest Air had the best reate with 80 % of its flights arriving on time. A test is conducted by randomly selecting 1010 Southwest flights and observing whether they arrive on time. (a) Find the probability that at least 44 flights arrive late.
Military radar and missile detection systems are designed to warn a country of an enemy attack. A reliability question is whether a detection system will be able to identify an attack and issue a warning. Assume that a particular detection system has a 0.80 probability of detecting a missile attack. Use the binomial probability distribution to answer the following questions. (a) What is the probability that a single detection system will detect an attack? 0.60 (b) If two detection systems are installed in the same area and operate independently, what is the probability that at least one of the systems will detect the attack? 0.84 (c) If three systems are installed, what is the probability that at least one of the systems will detect the attack? 0.936
Suppose you have data showing that 40% of tech firms saw their proportion of institutional ownership (IO) increase during the pandemic. When a tech firm’s IO increased, their total market capitalization (market cap) increased 70% of the time. Also, 25% of tech firms saw no market cap increase coupled with no increase in IO. a. What is the probability that a firm saw an increase in market cap if it saw no increase in IO? Provide your answer to the nearest 0.1%. b. Did an increase in IO and an increase in market cap have an effect on each other? Please explain your conclusion. c. What percentage of tech firms saw an increase in market cap but no increase in IO? Provide your answer to the nearest 0.1%.
A new automated production process averages 2.5 breakdowns per day. Because of the cost associated with a breakdown, management is concerned about the possibility of having three or more breakdowns during a day. Assume that breakdowns occur randomly, that the probability of a breakdown is the same for any two time intervals of equal length, and that breakdowns in one period are independent of breakdowns in other periods. What is the probability of having three or more breakdowns during a day? (Round your answer to four decimal places.)
A diagnostic test for disease X correctly identifies the disease 89% of the time. False positives occur 8%. It is estimated that 3.04% of the population suffers from disease X. Suppose the test is applied to a random individual from the population. Compute the following probabilities. (It may help to draw a probability tree.) The percentage chance that the test will be positive = % The probability that, given a positive result, the person has disease X =
If the probability is below 0.01, the data provide strong evidence that the null hypothesis is false. show how?
Many track hurdlers believe that they have a better chance of winning if they start in the inside lane that is closest to the field. For the data below, the lane closest to the field is Lane 1, the next lane is Lane 2, and so on until the outermost lane, Lane 6. The data lists the number of wins for track hurdlers in the different starting positions. Calculate the chi-square test statistic x to test the claim that the probabilities of winning are the same in the different positions. Use α = 0.05. The results are based on 240 wins. Starting Position 3 4 5 6 1 2 Number of Wins 44 32 33 50 36 45 O A. 6.750 O B. 12.592 O C. 15.541 O D. 9.326 C
A panel of meteorological and civil engineers studying emergency evacuation plans for Star Coast in the event of a hurricane has estimated that it would take between 13 and 18 hours to evacuate people leaving in low-lying land with the following probabilities: Probability 0.04 Time to Evacuate nearest hour 13 14 0.25 15 0.40 16 0.17 17 0.10 18 0.04 Obtain the standard deviation of the given probability distribution.
USA Today reported that Parkfield, California, is dubbed the world's earthquake capital because it sits on top of the notorious San Andreas fault. Since 1857, Parkfield has had a major earthquake on the average of 2.2 times every 22 years. (a) Explain why a Poisson probability distribution would be a good choice for r = number of earthquakes in a given time interval. select one Frequency of earthquakes is a rare occurrence. It is reasonable to assume the events are independent. Frequency of earthquakes is a common occurrence. It is reasonable to assume the events are independent. Frequency of earthquakes is a common occurrence. It is reasonable to assume the events are dependent. Frequency of earthquakes is a rare occurrence. It is reasonable to assume the events are dependent. (b) Compute the probability of at least one major earthquake in the next 22 years. Round ? to the nearest hundredth, and use a calculator. (Use 4 decimal places.)(c) Compute the probability that there…
One prominent physician claims that 70% of those with lung cancer are chain smokers. If his assertion is correct, (a) Find the: probability that of 10 such patients recently admitted to a hospital, fewer than half are chain smokers: (b) Find the probability that of 20 such patients recently admitted to a hospital, fewer than half are chain smokers.

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