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 Please read the Focus Problem at the beginning of this chapter. 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 (see the reference in this chapter's Focus Problem). 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 $r=46$ 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 $\alpha=0.01.$ 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 $1 \mathrm{s}$ ? 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. 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 7, Chapter 8: Understandable Statistics, Concepts and Methods
Uploaded: 2021-08-03T20:24:56-08:00
Duration: 7 min 15 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 Please read the Focus Problem at the beginning of this chapter. 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 (see the reference in this chapter's Focus Problem).
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 $r=46$ 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 $\alpha=0.01.$
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 $1 \mathrm{s}$ ? 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.
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 7

Instant Answer

Solved by Expert Carolyn Behr-Jerome

08/03/2021

Step 1

01$. The null hypothesis, denoted by $H_0$, is that the population proportion $p$ is equal to $0.301$. The alternate hypothesis, denoted by $H_a$, is that the population proportion $p$ is less than $0.301$. Show more…

Show all steps

Thanks for your feedback!

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 Please read the Focus Problem at the beginning of this chapter. 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 (see the reference in this chapter's Focus Problem). 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 $r=46$ 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 $\alpha=0.01.$ 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 $1 \mathrm{s}$ ? 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. 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?

Play audio

Feedback

Carolyn Behr-Jerome and 101 other subject Intro Stats / AP Statistics educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Decision_Rule

The decision rule in hypothesis testing is a systematic approach for deciding whether to reject the null hypothesis. It typically involves comparing the p-value to the predetermined significance level (?) or checking if the test statistic falls within a critical region. Based on the outcome of this comparison, researchers either reject or fail to reject the null hypothesis.

Type_I_Error_Misconception

A common misconception in hypothesis testing is the belief that rejecting the null hypothesis proves it to be false or that there is a direct probability (such as ?) that the decision was incorrect. In reality, the significance level only quantifies the probability of making a Type I error, meaning that if the null hypothesis is true, there is an ? chance of incorrectly rejecting it. It does not provide the probability that the hypothesis itself is false.

Contextual_Interpretation

Interpreting the results of a hypothesis test involves translating the statistical findings back into the real-world context of the study. This step requires considering the implications of the decision (rejecting or not rejecting H0), reflecting on the practical significance of the result, and articulating any conclusions or recommendations from the perspective of stakeholders or specific applications.

Test_Statistic

The test statistic is a standardized value computed from sample data that measures the degree of divergence from the null hypothesis. It is derived based on the sampling distribution and is used to decide whether to reject the null hypothesis. Common test statistics, such as the z-value, quantify how many standard errors the observed statistic is away from the hypothesized population parameter.

P_value

The p-value quantifies the probability of obtaining a sample test statistic as extreme as, or more extreme than, the one observed, assuming that the null hypothesis is true. It provides a measure of the strength of evidence against the null hypothesis. A small p-value indicates that the observed result is highly unlikely under the null hypothesis, thus favoring the alternative hypothesis.

Sampling_Distribution

The sampling distribution refers to the probability distribution of a statistical measure (like the sample proportion or sample mean) that is formed when repeatedly sampling from the population. In hypothesis testing, understanding the appropriate sampling distribution—such as the normal or z-distribution for large samples—is vital, as it underpins the computation of the test statistic and the subsequent p-value.

Null_and_Alternative_Hypotheses

The null hypothesis (H0) typically represents a statement of no effect or a default claim about a population parameter, while the alternative hypothesis (Ha) is the claim that contradicts H0 and is accepted if the evidence in the sample data is sufficiently strong. Formulating these hypotheses clearly is essential, as they guide the testing procedure and the interpretation of the results.

Significance_Level

The significance level, denoted by ?, is a threshold probability used in hypothesis testing to determine whether the observed results are statistically unusual under the assumption that the null hypothesis is true. It represents the maximum acceptable probability of committing a Type I error, which is rejecting a true null hypothesis, and serves as a benchmark for decision making in tests of statistical significance.

Sample_Size_Considerations

Determining if the sample size is sufficiently large is crucial for the validity of the approximations used in hypothesis testing. Many tests, especially those based on the Central Limit Theorem, assume that the sample size is large enough for the sampling distribution of the sample statistic to be approximately normal, thereby justifying the use of z or t statistics.

Please give Ace some feedback