Question

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain. Compute the value of the sample test statistic and corresponding $z$ value. (c) Find the $P$-value of the test statistic. 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 $\alpha ?$ (e) Interpret your conclusion in the context of the application. Focus Problem: Benford's Law Again, suppose you are the auditor for a very large corporation. The revenue file contains millions of numbers in a large computer data bank (see Problem 7). You draw a random sample of $n=228$ numbers from this file and $r=92$ have a first nonzero digit of $1 .$ Let $p$ represent the population proportion of all numbers in the computer file that have a leading digit of $1 .$ i. Test the claim that $p$ is more than $0.301 .$ Use $\alpha=0.01.$ ii. If $p$ is in fact larger than $0.301,$ it would seem there are too many numbers in the file with leading Is. Could this indicate that the books have been "cooked" by artificially lowering numbers in the file? Comment from the point of view of the Internal Revenue Service. Comment from the perspective of the Federal Bureau of Investigation as it looks for "profit skimming" by unscrupulous employees. iii. Comment on the following statement: "If we reject the null hypothesis at level of significance $\alpha,$ we have not proved $H_{0}$ to be false. We can say that the probability is $\alpha$ that we made a mistake in rejecting $H_{0}$ " Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?

Name: Problem 8, Chapter 8: Understandable Statistics, Concepts and Methods
Uploaded: 2021-08-03T20:36:40-08:00
Duration: 6 min 54 s
Channel: Carolyn Behr-Jerome

Please provide the following information.
(a) What is the level of significance? State the null and alternate hypotheses.
(b) Check Requirements What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain. Compute the value of the sample test statistic and corresponding $z$ value.
(c) Find the $P$-value of the test statistic. 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 $\alpha ?$
(e) Interpret your conclusion in the context of the application.
Focus Problem: Benford's Law Again, suppose you are the auditor for a very large corporation. The revenue file contains millions of numbers in a large computer data bank (see Problem 7). You draw a random sample of $n=228$ numbers from this file and $r=92$ have a first nonzero digit of $1 .$ Let $p$ represent the population proportion of all numbers in the computer file that have a leading digit of $1 .$
i. Test the claim that $p$ is more than $0.301 .$ Use $\alpha=0.01.$
ii. If $p$ is in fact larger than $0.301,$ it would seem there are too many numbers in the file with leading Is. Could this indicate that the books have been "cooked" by artificially lowering numbers in the file? Comment from the point of view of the Internal Revenue Service. Comment from the perspective of the Federal Bureau of Investigation as it looks for "profit skimming" by unscrupulous employees.
iii. Comment on the following statement: "If we reject the null hypothesis at level of significance $\alpha,$ we have not proved $H_{0}$ to be false. We can say that the probability is $\alpha$ that we made a mistake in rejecting $H_{0}$ " Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?

Understandable Statistics, Concepts and Methods

Charles Henry Brase,… 12th Edition

Chapter 8, Problem 8

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Solved by Expert Carolyn Behr-Jerome

08/03/2021

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01$. The null hypothesis is $H_{0}: p=0.301$ and the alternate hypothesis is $H_{1}: p>0.301$. Show more…

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Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? Do you think the sample size is sufficiently large? Explain. Compute the value of the sample test statistic and corresponding $z$ value. (c) Find the $P$-value of the test statistic. 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 $\alpha ?$ (e) Interpret your conclusion in the context of the application. Focus Problem: Benford's Law Again, suppose you are the auditor for a very large corporation. The revenue file contains millions of numbers in a large computer data bank (see Problem 7). You draw a random sample of $n=228$ numbers from this file and $r=92$ have a first nonzero digit of $1 .$ Let $p$ represent the population proportion of all numbers in the computer file that have a leading digit of $1 .$ i. Test the claim that $p$ is more than $0.301 .$ Use $\alpha=0.01.$ ii. If $p$ is in fact larger than $0.301,$ it would seem there are too many numbers in the file with leading Is. Could this indicate that the books have been "cooked" by artificially lowering numbers in the file? Comment from the point of view of the Internal Revenue Service. Comment from the perspective of the Federal Bureau of Investigation as it looks for "profit skimming" by unscrupulous employees. iii. Comment on the following statement: "If we reject the null hypothesis at level of significance $\alpha,$ we have not proved $H_{0}$ to be false. We can say that the probability is $\alpha$ that we made a mistake in rejecting $H_{0}$ " Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?

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Key Concepts

Hypothesis Testing Framework

This concept involves setting up a formal procedure to decide whether sample data provide sufficient evidence to reject a null hypothesis in favor of an alternative hypothesis. It includes determining the significance level, computing a test statistic based on the sampling distribution, calculating the corresponding P-value, and making a decision about the null hypothesis. Overall, it provides a systematic method for making inferences about population parameters based on sample data.

Level of Significance

The level of significance, often denoted by ?, is the criterion used by the researcher to decide whether to reject the null hypothesis. It represents the probability of making a Type I error, that is, rejecting a true null hypothesis. It sets the acceptable threshold for the risk of false positives in hypothesis testing.

Null Hypothesis and Alternate Hypothesis

These are mutually exclusive statements that define the two contrasting claims in hypothesis testing. The null hypothesis (H0) generally states that there is no effect or no difference, serving as the default assumption, while the alternative hypothesis (Ha) asserts the presence of an effect or a difference. Formulating clear hypotheses is essential for framing the statistical test and subsequent decision-making.

Sampling Distribution

This concept refers to the probability distribution of a given statistic based on a random sample drawn from the population. In hypothesis testing, particularly in proportion tests, the sampling distribution is typically approximated by a normal distribution under the Central Limit Theorem when the sample size is large enough. It is crucial for determining the likelihood of observing the sample statistic under the null hypothesis.

Sample Size Adequacy

Ensuring that the sample size is sufficiently large is a key requirement in many statistical tests, as it allows the assumption of a normal distribution for the sampling distribution of the test statistic. Adequate sample size helps to justify the use of the Central Limit Theorem and ensures reliable inference by reducing the sampling error.

Test Statistic and Z Value

The test statistic is a standardized value calculated from sample data, used to assess the evidence against the null hypothesis. In tests involving proportions, a z statistic is often computed by comparing the observed sample proportion to the hypothesized proportion, divided by the standard error. This standardization enables referencing the standard normal distribution for decision-making.

P-value and Decision Rule

The P-value is the probability, under the null hypothesis, of obtaining a test statistic as extreme as or more extreme than the observed value. It is used to quantify the strength of evidence against the null hypothesis. The decision rule in hypothesis testing involves comparing the P-value to the predetermined level of significance (?); if the P-value is less than ?, the null hypothesis is rejected.

Contextual Interpretation

Beyond the numerical outcomes, interpreting the test results in the context of the application is essential. This includes understanding the practical implications of rejecting or not rejecting the null hypothesis in real-world situations, such as in audits, fraud detection, or other applied fields. Contextual interpretation links statistical conclusions with domain-specific insights and potential actions.

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