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Chapter 11: Q11-2BSC (page 533)

Cybersecurity When using the data from Exercise 1 to test for goodness-of-fit with the distribution described by Benford’s law, identify the null and alternative hypotheses.

Short Answer

Expert verified

The null hypothesis and the alternative hypothesis is

\(\begin{aligned}{l}{H_0}:{p_1} = 0.301,{p_2} = 0.176,{p_3} = 0.125,{p_4} = 0.097,{p_5} = 0.079,{p_6} = 0.067,{p_7} = 0.058,\\\;\;\;\;\;{p_8} = 0.051,{p_9} = 0.046\end{aligned}\)

\({H_1}:\)At least one of the proportions will differ from the others.

Step by step solution

01

Given information

The observed frequencies and the expected frequencies of the leading digits of inter-arrival traffic times are tabulated.

02

Identify the hypotheses

The following hypotheses are set up to test for the goodness of fit test of the given distribution:

Null Hypothesis:

The null hypothesis is that in which the proportions of all the leading digits should be equal to the claimed value.

\(\begin{aligned}{l}{H_0}:{p_1} = 0.301,{p_2} = 0.176,{p_3} = 0.125,{p_4} = 0.097,{p_5} = 0.079,{p_6} = 0.067,{p_7} = 0.058,\\\;\;\;\;\;{p_8} = 0.051,{p_9} = 0.046\end{aligned}\)

Alternative Hypothesis:

The alternative hypothesis is that in which at least one of the proportions should not be equal to the claimed value.

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Most popular questions from this chapter

Benford’s Law. According to Benford’s law, a variety of different data sets include numbers with leading (first) digits that follow the distribution shown in the table below. In Exercises 21–24, test for goodness-of-fit with the distribution described by Benford’s law.

Leading Digits

Benford's Law: Distributuon of leading digits

1

30.10%

2

17.60%

3

12.50%

4

9.70%

5

7.90%

6

6.70%

7

5.80%

8

5.10%

9

4.60%

Tax Cheating? Frequencies of leading digits from IRS tax files are 152, 89, 63, 48, 39, 40, 28, 25, and 27 (corresponding to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively, based on data from Mark Nigrini, who provides software for Benford data analysis). Using a 0.05 significance level, test for goodness-of-fit with Benford’s law. Does it appear that the tax entries are legitimate?

In soccer, serious fouls in the penalty box result in a penalty kick withone kicker and one defending goalkeeper. The table below summarizes results from 286 kicksduring games among top teams (based on data from “Action Bias Among Elite Soccer Goalkeepers:

The Case of Penalty Kicks,” by Bar-Eli et al., Journal of Economic Psychology,Vol.28, No. 5). In the table, jump direction indicates which way the goalkeeper jumped, where thekick direction is from the perspective of the goalkeeper. Use a 0.05 significance level to test theclaim that the direction of the kick is independent of the direction of the goalkeeper jump. Dothe results support the theory that because the kicks are so fast, goalkeepers have no time toreact, so the directions of their jumps are independent of the directions of the kicks?

Goalkeeper Jump

Left

Center

Right

Kick to Left

54

1

37

Kick to Center

41

10

31

Kick to Right

46

7

59

In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and, or critical value, and state the conclusion.

Kentucky Derby The table below lists the frequency of wins for different post positions through the 141st running of the Kentucky Derby horse race. A post position of 1 is closest to the inside rail, so the horse in that position has the shortest distance to run. (Because the number of horses varies from year to year, only the first 10 post positions are included.) Use a 0.05 significance level to test the claim that the likelihood of winning is the same for the different post positions. Based on the result, should bettors consider the post position of a horse racing in the Kentucky Derby?

Post Position

1

2

3

4

5

6

7

8

9

10

Wins

19

14

11

15

15

7

8

12

5

11

Cybersecurity The accompanying Statdisk results shown in the margin are obtained from the data given in Exercise 1. What should be concluded when testing the claim that the leading digits have a distribution that fits well with Benford’s law?

Winning team data were collected for teams in different sports, with the results given in the table on the top of the next page (based on data from “Predicting Professional Sports Game Outcomes fromIntermediateGame Scores,” by Copper, DeNeve, and Mosteller, Chance,Vol. 5, No. 3–4). Use a 0.10significance level to test the claim that home/visitor wins are independent of the sport. Given that among the four sports included here, baseball is the only sport in which the home team canmodify field dimensions to favor its own players, does it appear that baseball teams are effective in using this advantage?

Basketball

Baseball

Hockey

Football

Home Team Wins

127

53

50

57

Visiting Team Wins

71

47

43

42
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