Question

Recall that Benford's Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Now suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say you took a random sample of n = 215 numerical entries from the file and r = 49 of the entries had a first nonzero digit of 1. Let p represent the population proportion of all numbers in the corporate file that have a first nonzero digit of 1.

(i) Test the claim that p is less than 0.301. Use α = 0.05.(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p = 0.301; H₁: p ≠ 0.301H₀: p < 0.301; H₁: p = 0.301    H₀: p = 0.301; H₁: p < 0.301H₀: p = 0.301; H₁: p > 0.301

(b) What sampling distribution will you use?

The standard normal, since np > 5 and nq > 5.The Student's t, since np > 5 and nq > 5.    The standard normal, since np < 5 and nq < 5.The Student's t, since np < 5 and nq < 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.)


(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)


Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.05 level to conclude that the true proportion of numbers with a leading 1 in the revenue file is less than 0.301.There is insufficient evidence at the 0.05 level to conclude that the true proportion of numbers with a leading 1 in the revenue file is less than 0.301.    

(ii) If p is in fact less than 0.301, would it make you suspect that there are not enough numbers in the data file with leading 1's? Could this indicate that the books have been "cooked" by "pumping up" or inflating the numbers? Comment from the viewpoint of a stockholder. Comment from the perspective of the Federal Bureau of Investigation as it looks for money laundering in the form of false profits.

Yes. The revenue data file does not seem to include more numbers with higher first nonzero digits than Benford's law predicts.No. The revenue data file seems to include more numbers with higher first nonzero digits than Benford's law predicts.    Yes. The revenue data file seems to include more numbers with higher first nonzero digits than Benford's law predicts.No. The revenue data file does not seem to include more numbers with higher first nonzero digits than Benford's law predicts.

(iii) Comment on the following statement: If we reject the null hypothesis at level of significance α, we have not proved H_o to be false. We can say that the probability is α that we made a mistake in rejecting H_o. Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?

We have not proved H₀ to be false. Because our data lead us to accept the null hypothesis, more investigation is not merited.We have not proved H₀ to be false. Because our data lead us to reject the null hypothesis, more investigation is merited.    We have proved H₀ to be false. Because our data lead us to reject the null hypothesis, more investigation is not merited.We have not proved H₀ to be false. Because our data lead us to reject the null hypothesis, more investigation is not merited.

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2.

[–/8 Points]DETAILSBBUNDERSTAT12 8.3.009.

MY NOTES

Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years†. Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 35 arrests last month, 26 were of males aged 15 to 34 years. Use a 10% level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%.
(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p ≠ 0.7; H₁: p = 0.7H₀: p = 0 .7; H₁: p < 0.7    H₀: p = 0.7; H₁: p > 0.7H₀: p < 0 .7; H₁: p = 0.7H₀: p = 0.7; H₁: p ≠ 0.7

(b) What sampling distribution will you use?

The standard normal, since np > 5 and nq > 5.The standard normal, since np < 5 and nq < 5.    The Student's t, since np < 5 and nq < 5.The Student's t, since np > 5 and nq > 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.)


(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)


Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.10 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.10 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.There is insufficient evidence at the 0.10 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.    

3.

[–/8 Points]DETAILSBBUNDERSTAT12 8.3.010.

MY NOTES

Women athletes at the a certain university have a long-term graduation rate of 67%. Over the past several years, a random sample of 36 women athletes at the school showed that 21 eventually graduated. Does this indicate that the population proportion of women athletes who graduate from the university is now less than 67%? Use a 10% level of significance.(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p < 0.67; H₁: p = 0.67H₀: p = 0.67; H₁: p > 0.67    H₀: p = 0.67; H₁: p ≠ 0.67H₀: p = 0.67; H₁: p < 0.67

(b) What sampling distribution will you use?

The standard normal, since np < 5 and nq < 5.The Student's t, since np > 5 and nq > 5.    The Student's t, since np < 5 and nq < 5.The standard normal, since np > 5 and nq > 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.)


(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)


Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.10 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.10 level to conclude that the true proportion of women athletes who graduate is less than 0.67.There is insufficient evidence at the 0.10 level to conclude that the true proportion of women athletes who graduate is less than 0.67.    

Answer 1

1)

Ho :   p =    0.301                  
H1 :   p <   0.301       (Left tail test)         
                          

The standard normal, since np > 5 and nq > 5.

Level of Significance,   α =    0.05                  
Number of Items of Interest,   x =   49                  
Sample Size,   n =    215                  
                          
Sample Proportion ,    p̂ = x/n =    0.2279                  
                          
Standard Error ,    SE = √( p(1-p)/n ) =    0.0313                  
Z Test Statistic = ( p̂-p)/SE = (   0.2279   -   0.301   ) /   0.0313   =   -2.34
                          

p-Value   =   0.0097 [excel function =NORMSDIST(z)]              
Decision:   p-value<α , reject null hypothesis

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant                      
There is sufficient evidence at the 0.05 level to conclude that the true proportion of numbers with a leading 1 in the revenue file is less than 0.301.                   

We have proved H0 to be false. Because our data lead us to reject the null hypothesis, more investigation is not merited.

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