91Ó°ÊÓ

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 5). 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 1's. 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?

Step by step solution

01

Define Hypotheses

To test the claim that \( p > 0.301 \), we set up the hypotheses as follows. Null hypothesis \( H_0: p \leq 0.301 \).Alternative hypothesis \( H_a: p > 0.301 \). This is a right-tailed test.

02

Calculate Sample Proportion

The sample proportion \( \hat{p} \) can be calculated by dividing the number of successes \( r \) by the sample size \( n \). \[ \hat{p} = \frac{92}{228} \approx 0.4035 \]

03

Compute Test Statistic

For a proportion, the test statistic \( z \) is computed using the formula: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \] where \( p_0 = 0.301 \) is the hypothesized population proportion. \[ z = \frac{0.4035 - 0.301}{\sqrt{\frac{0.301 \times 0.699}{228}}} \approx 3.470 \]

04

Determine Critical Value and Decision

At \( \alpha = 0.01 \) for a right-tailed test, find the critical value from the standard normal distribution table. The critical value for \( z \) at \( \alpha = 0.01 \) is approximately 2.33. Since \( z \approx 3.470 \) is greater than 2.33, we reject \( H_0 \).

05

Interpret Results

Since \( H_0 \) is rejected, there is statistical evidence to support that \( p > 0.301 \). This may indicate an atypically high number of leading "1" digits, suggesting potential manipulation or data tampering. From the IRS perspective, it could be significant taxation implications, and from the FBI's point of view, possible "profit skimming."

06

Evaluate the Statement and Recommendations

The statement about not proving \( H_0 \) false if rejected is correct; hypothesis tests do not prove hypotheses but evaluate evidence based on a chosen significance level. With \( \alpha = 0.01 \), there's a 1% chance of incorrectly rejecting \( H_0 \). Given the test outcome, further investigation is recommended before making strong accusations of fraud.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Proportion

In statistics, the population proportion, denoted as \( p \), represents the fraction of the total population that possesses a particular characteristic. In the context of our exercise, we are interested in the proportion of numbers in a computer data file that have a leading digit of "1". This proportion is what we're trying to understand better by examining a sample from the population.

The population proportion is commonly unknown in practical scenarios, and thus, we rely on estimates from sample data to make inferences. By drawing a random sample of 228 numbers and observing that 92 of these have "1" as the leading digit, we calculate the sample proportion (denoted as \( \hat{p} \)) as \( \hat{p} = \frac{92}{228} \approx 0.4035 \). This sample proportion serves as a point estimate for the true population proportion \( p \).

Understanding population proportion is crucial for forming hypotheses about the population, which leads us to conduct hypothesis testing to decide if our data provides enough evidence to support a statistical claim.

Right-Tailed Test

A right-tailed test is a type of hypothesis test where the alternative hypothesis states that the parameter of interest is greater than the value specified in the null hypothesis. In this exercise, we aim to test if the population proportion \( p \) of numbers with a leading digit of "1" is greater than 0.301.

The hypotheses are set up as follows:

• Null hypothesis \( H_0: p \leq 0.301 \)
• Alternative hypothesis \( H_a: p > 0.301 \)

The right-tailed test focuses on the upper end of the distribution. We compute a test statistic \( z \), which indicates how far the observed sample proportion is from the hypothesized proportion, based on the standard normal distribution. If this computed \( z \)-value is greater than the critical value from the normal distribution table for the chosen significance level, we reject the null hypothesis \( H_0 \). In our scenario, the calculated \( z \)-value was 3.470, which exceeded the critical value of 2.33, providing evidence that \( p \) is indeed greater than 0.301.

Standard Normal Distribution

The standard normal distribution is a special type of normal distribution with a mean of 0 and a standard deviation of 1. It is denoted by \( Z \) and is used extensively in statistics to standardize individual scores so they can be easily compared across different normal distributions.

In hypothesis testing, the test statistic \( z \) is often computed by applying this standardization process. This process involves transforming the observed sample data into a format that can be easily compared to the critical values of the standard normal distribution. The formula used to find the test statistic for proportions is:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]where \( \hat{p} \) is the observed sample proportion, \( p_0 \) is the hypothesized population proportion, and \( n \) is the sample size.

By comparing this calculated \( z \)-value to the critical \( z \)-value for our significance level, we can make decisions about whether or not to reject the null hypothesis, thereby determining the statistical validity of our claims.

Significance Level

The significance level, denoted \( \alpha \), represents the threshold at which we decide whether to reject the null hypothesis. It reflects the probability of making a type I error, which occurs when we mistakenly reject a true null hypothesis.

In our exercise, the significance level is set at \( \alpha = 0.01 \), implying a 1% risk of a type I error. This means we are willing to accept only a 1% chance of concluding that the population proportion \( p \) is greater than 0.301 when it is not.

The choice of \( \alpha \) is important as it influences how stringently we test the hypothesis. A lower \( \alpha \) (like 0.01) necessitates stronger evidence to reject \( H_0 \). In our test, the calculated \( z \)-value of 3.470 is compared to the critical \( z \)-value corresponding to \( \alpha = 0.01 \), which is 2.33. Since our \( z \)-value is greater, we reject \( H_0 \) and conclude that there is sufficient evidence to support that \( p > 0.301 \).

Significance levels are pivotal in maintaining a balance between the risks of making incorrect conclusions while also allowing us to make informed inferences about population parameters.