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 = 217 numerical entries from the file and r = 47 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 ?? = 0.05. (a) What is the level of significance? .05 State the null and alternate hypotheses. H0: p = 0.301; H1: p < 0.301 H0: p = 0.301; H1: p > 0.301 H0: p = 0.301; H1: p ? 0.301 H0: 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 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.) 2.70 (c) Find the P-value of the test statistic. (Round your answer to four decimal places.)
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301. The level of significance is given as 0.05. (a) The null and alternate hypotheses are: $$ H_0: p = 0.301 \\ H_1: p < 0.301 $$ (b) We will use the standard normal distribution for the sampling distribution since np > 5 and nq > 5. In this case, n = 217, p = Show more…
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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. 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 = 247 numerical entries from the file and r = 60 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. Test the claim that p is less than 0.301 by using α = 0.1. Are the data statistically significant at the significance level? Based on your answers, will you reject or fail to reject the null hypothesis? a. The P-value is less than the level of significance so the data are statistically significant. Thus, we reject the null hypothesis. b. The P-value is less than the level of significance so the data are statistically significant. Thus, we reject the null hypothesis. c. The P-value is less than the level of significance so the data are statistically significant. Thus, we fail to reject the null hypothesis. d. The P-value is less than the level of significance so the data are not statistically significant. Thus, we reject the null hypothesis. e. The P-value is greater than the level of significance so the data are not statistically significant. Thus, we reject the null hypothesis.
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Lucky numbers If people choose lottery numbers at random, the last digit should be equally likely to be any of the ten digits from 0 to $9 .$ Let $p$ measure the proportion of choices that end with the digit 7. If choices are random, we would expect $p=0.10$, but if people have a special preference for numbers ending in 7 the proportion will be greater than 0.10 . Suppose that we test this by asking a random sample of 20 people to give a three-digit lottery number and find that four of the numbers have 7 as the last digit. Figure 5.13 shows a randomization distribution of proportions for 5000 simulated samples under the null hypothesis $H_{0}: p=0.10$.(a) Use the sample proportion $\hat{p}=0.20$ and a stan- (c) Compare the p-value obtained from the nordard error estimated from the randomization $\quad$ mal distribution in part (b) to the p-value shown distribution to compute a standardized test $\quad$ for the randomization distribution. Explain why statistic. $\quad$ there might be a discrepancy between these two (b) Use the normal distribution to find a p-value for values. an upper tail alternative based on the test statistic found in part (a).
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