Chapter 9.3, Problem 8P

For Problems 7-21, please provide the following information.

(a) What is the level of significance? Stale 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 sour answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistic ally significant at level a?

(c) 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 1s. 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 a 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 ?

To determine

The level of significance, null and alternative hypothesis.

Answer to Problem 8P

Solution: The level of significance is $\alpha =0.01$. The null hypothesis is $H_0:p=0.301$ and alternative hypothesis $H_A:p>0.301$.

Explanation of Solution

The level of significance is defined as the probability of rejecting the null hypothesis when it is true, it is denoted by $\alpha =0.01$.

Null hypothesis $H_0:p=0.301$

Alternative hypothesis $H_A:p>0.301$

To determine

To find: The sampling distribution that should be used and compute the z value of the sample test statistic.

Answer to Problem 8P

Solution: The sampling distribution $\overline{x}$ is normal. The sample test statistic, z is 3.37.

Explanation of Solution

Calculation:

The $\widehat{p}$ distribution is approximately normal, with mean $p$ and standard deviation $\sqrt{\frac{pq}{n}}$.

$\begin{array}{l}p=0.301,q=1-p=0.699\\ \widehat{p}=\frac{r}{n}\\ \widehat{p}=\frac{92}{228}=0.404\end{array}$

The standardized sample test statistic for $\widehat{p}$ is

$\begin{array}{l}z=\frac{(\widehat{p}-p)}{\sqrt{\frac{pq}{n}}}\\ z=\frac{(0.404-0.301)}{\sqrt{\frac{0.301(1-0.301)}{228}}}\\ z=3.374\\ z\approx 3.37\end{array}$

To determine

To find: The P-value of the test statistic and sketch the sampling distribution showing the area corresponding to the P-value.

Answer to Problem 8P

Solution: The P-value of the test statistic is 0.0004.

Explanation of Solution

Calculation:

We have z = 3.37

$P-\text{value}\text{=}P(z>3.37)=1-P(z<3.37)$

Using Table 3 from the Appendix to find the specified area:

$P-\text{value}\text{=}1-0.9996=0.0004$

Thus P- value is 0.0004.

Graph:

To draw the required graphs using the Minitab, follow the below instructions:

Step 1: Go to the Minitab software.

Step 2: Go to Graph > Probability distribution plot > View probability.

Step 3: Select ‘Normal’ and enter Mean 0 and Standard deviation 1.

Step 4: Click on the Shaded area > X value.

Step 5: Enter X-value as 3.37 and select ‘Right tail’.

Step 6: Click on OK.

The obtained distribution graph is:

$P-\text{value}\approx 0.0004$

To determine

Whether we reject or fail to reject the null hypothesisand whether the data is statistically significant for a level of significance of 0.01.

Answer to Problem 8P

Solution: The P-value $<\alpha$, hence we have to reject the $H_0$. The data is statistically significant for a level of significance of 0.01.

Explanation of Solution

The P-value of 0.0004 is less than the level of significance ($\alpha$) of 0.01. Therefore we have enough evidence to reject the null hypothesis $H_0$, hence the data is statistically significant for a level of significance of 0.01.

To determine

The interpretation for the conclusion.

Answer to Problem 8P

Solution: There is sufficient evidence to conclude that population proportion of numbers with leading “1” in the revenue file is more than the probability 0.301.

Explanation of Solution

The P-value of 0.0004 is less than the level of significance ($\alpha$) of 0.01. Therefore we have enough evidence to reject the null hypothesis $H_0$. There is sufficient evidence to conclude that population proportion of numbers with leading “1” in the revenue file is more than the probability 0.301.

To determine

To explain: Whether it is suspect that there are too many numbers in the data file with leading 1's.

Answer to Problem 8P

Solution: Yes. The revenue data file seems to be too many entries with leading digit 1. 

Explanation of Solution

There are too many numbers in the data file with leading 1's. So, we cannot say that it is an indication of the books have been “cooked” by artificially lowering numbers in the file. From the viewpoint of the Internal Revenue Service and the Federal Bureau of Investigation as it looks for “profit skimming”, it may be true or false because there are too many numbers in the data file with leading 1’s.

To determine

To explain: Whether it recommends further investigation before accusing the company of fraud.

Answer to Problem 8P

Solution: Our data lead us to reject the null hypothesis, more investigation is merited. 

Explanation of Solution

Since, we reject the null hypothesis $H_0$ at level of significance 0.01 but we have not proved $H_0$ to be false. Because our data lead us to reject the null hypothesis and we conclude that there are too many enough numbers in the data file with leading 1's. Hence, we recommend further investigation before accusing the company of fraud because more investigation will give better results.