/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 As an example of a situation in ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 5

As an example of a situation in which several different statistics could reasonably be used to calculate a point estimate, consider a population of \(N\) invoices. Associated with each invoice is its "book value," the recorded amount of that invoice. Let \(T\) denote the total book value, a known amount. Some of these book values are erroneous. An audit will be carried out by randomly selecting \(n\) invoices and determining the audited (correct) value for each one. Suppose that the sample gives the following results (in dollars). \begin{tabular}{lrcrrr} \hline & \multicolumn{5}{c}{ Invoice } \\ \cline { 2 - 6 } & \(\mathbf{1}\) & \(\mathbf{2}\) & \(\mathbf{3}\) & \(\mathbf{4}\) & \(\mathbf{5}\) \\ \hline Book value & 300 & 720 & 526 & 200 & 127 \\ Audited value & 300 & 520 & 526 & 200 & 157 \\ Error & 0 & 200 & 0 & 0 & \(-30\) \\ \hline \end{tabular} Let \(\bar{X}=\) the sample mean audited value, \(\bar{Y}=\) the sample mean book value, and \(\bar{D}=\) the sample mean error. Propose three different statistics for estimating the total audited (i.e. correct) value \(\theta\) - one involving just \(N\) and \(\bar{X}\), another involving \(N, T\), and \(\bar{D}\), and the last involving \(T\) and \(\bar{X} / \bar{Y}\). Then calculate the resulting estimates when \(N=5,000\) and \(T=1,761,300\) (The article "Statistical Models and Analysis in Auditing,", Statistical Science, 1989: 2 - 33 discusses properties of these estimators).

Short Answer

Expert verified
Estimates: \(\theta_1 = 1,703,000\), \(\theta_2 = 1,591,300\), \(\theta_3 \approx 1,601,500\) dollars.

Step by step solution

01

Calculate the Sample Mean Audited Value

First, let's determine the sample mean of the audited values, \(\bar{X}\). We add up all the audited values and divide by the number of invoices (5): \[ \bar{X} = \frac{300 + 520 + 526 + 200 + 157}{5} = \frac{1703}{5} = 340.6 \] dollars.
02

Calculate the Sample Mean Book Value

Next, calculate the sample mean of the book values, \(\bar{Y}\). Add all the book values and divide by the number of invoices (5): \[ \bar{Y} = \frac{300 + 720 + 526 + 200 + 127}{5} = \frac{1873}{5} = 374.6 \] dollars.
03

Calculate the Sample Mean Error

Determine the sample mean error, \(\bar{D}\), by summing the errors and dividing by the number of invoices (5): \[ \bar{D} = \frac{0 + 200 + 0 + 0 - 30}{5} = \frac{170}{5} = 34 \] dollars.
04

Estimate Total Audited Value using \(N\) and \(\bar{X}\)

Using \(N = 5000\) and \(\bar{X} = 340.6\), the estimate of the total audited value \(\theta_1\) is:\[ \theta_1 = N \cdot \bar{X} = 5000 \cdot 340.6 = 1,703,000 \] dollars.
05

Estimate Total Audited Value using \(N\), \(T\), and \(\bar{D}\)

Using \(N = 5000\), total book value \(T = 1,761,300\), and \(\bar{D} = 34\), the estimate of the total audited value \(\theta_2\) is:\[ \theta_2 = T - N \cdot \bar{D} = 1,761,300 - 5000 \cdot 34 = 1,591,300 \] dollars.
06

Estimate Total Audited Value using \(T\) and \(\frac{\bar{X}}{\bar{Y}}\)

Using \(T = 1,761,300\), \(\bar{X} = 340.6\), and \(\bar{Y} = 374.6\), the estimate of the total audited value \(\theta_3\) is:\[ \theta_3 = T \cdot \frac{\bar{X}}{\bar{Y}} = 1,761,300 \cdot \frac{340.6}{374.6} \approx 1,601,500 \] dollars.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Sample Mean
In statistics, the sample mean is a vital, straightforward concept frequently used for estimation. The sample mean, denoted as \( \bar{X} \), is the arithmetic average of a set of values. It provides a central value for the data and serves as an estimator of the population mean. In our example, the sample mean of the audited values is calculated by summing all the audited values and dividing by the number of items, in this case, invoices.
To calculate the sample mean, the formula is:
\[ \bar{X} = \frac{\sum_{i=1}^{n} X_i}{n} \]
  • Sum all the observed values: 300, 520, 526, 200, and 157.
  • Divide by the number of observations: 5.
  • The result is \( \bar{X} = 340.6 \) dollars for our sample.
This mean value helps in estimating other quantities like the total audited value of all invoices when multiplied by the total number of items (\(N\)) in the entire population.
Statistics in Auditing
Statistics play a crucial role in auditing by providing methodologies to assess and verify financial records. In auditing, statistical methods help in analyzing sample data to form conclusions about a larger dataset or population. Auditors use sampling to predict potential errors and estimate the total audited value, which influences decision-making regarding financial statements.
Auditing strategies are guided by:
  • Identification of risks in financial records.
  • Determining the sample size for analysis to ensure reliability.
  • Using statistical models to estimate the total correct or audited value of financial items.
  • Performing error analysis to detect discrepancies.
These strategies enhance the efficiency and accuracy of audits, enabling informed recommendations to stakeholders. In the context of our exercise, we use various sample statistics to estimate the total audited value from a selected invoice sample. This involves calculating both sample means and sample errors and applying these findings to the entire population of invoices.
Total Audited Value Estimation
Estimating the total audited value, or the correct value, involves using different statistical techniques to extrapolate sample findings to a broader dataset. In our exercise, we explore three main methods to estimate this value, denoted as \( \theta \).

The approaches include:
  • Using the sample mean audited value \(\bar{X}\): Multiply \(\bar{X}\) by the total number of invoices \(N\).
    This gives \( \theta_1 = N \cdot \bar{X} = 1,703,000 \) dollars.

  • Using total book value \( T \) and sample mean error \( \bar{D} \):
    The formula is \( \theta_2 = T - N \cdot \bar{D} = 1,591,300 \) dollars. This method adjusts the total book value based on average sample errors.

  • Applying a ratio method with \( \frac{\bar{X}}{\bar{Y}} \):
    Calculated as \( \theta_3 = T \cdot \frac{\bar{X}}{\bar{Y}} = 1,601,500 \) dollars.
    This scales the total book value using the ratio of the means.
These methods illustrate the variability in estimation techniques, depending on available data and required accuracy.
Error Analysis
Error analysis is an essential component of statistical auditing. It involves identifying and evaluating discrepancies between recorded and audited values. The objective is to comprehend and quantify errors to adjust estimates and improve accuracy.
In our illustration:
  • We calculate the sample mean error \( \bar{D} \) by averaging the deviations between book values and audited values.
  • Using \( \bar{D} \) helps assess systematic errors across the population, providing a basis for adjusting the total audited value estimation.
    For instance, method two uses \( \bar{D} \) in conjunction with the total book value to deliver a corrected total estimate.
Error analysis not only assists in refining total value estimates but also aids in understanding the variability and potential biases within the data. Robust error analysis ensures a deeper grasp of financial data, supporting accurate audit judgments.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The long run proportion of vehicles that pass a certain emissions test is \(p\). Suppose that three vehicles are independently selected for testing. Let \(X_{i}=1\) if the \(i\) th vehicle passes the test and \(X_{i}=0\) otherwise \((i=1,2,3)\), and let \(X=X_{1}+\) \(X_{2}+X_{3}\). Use the definition of sufficiency to show that \(X\) is sufficient for \(p\) by obtaining the conditional distribution of the \(X_{i}\) 's given that \(X=x\) for each possible value \(x\). Then generalize by giving an analogous argument for the case of \(n\) vehicles.

At time \(t=0,20\) identical components are put on test. The lifetime distribution of each is exponential with parameter \(\lambda\). The experimenter then leaves the test facility unmonitored. On his return 24 h later, the experimenter immediately terminates the test after noticing that \(y=15\) of the 20 components are still in operation (so 5 have failed). Derive the mle of \(\lambda\). [Hint: Let \(Y=\) the number that survive \(24 \mathrm{~h}\). Then \(Y \sim \operatorname{Bin}(n, p)\). What is the mle of \(p\) ? Now notice that \(p=P\left(X_{i} \geq 24\right)\), where \(X_{i}\) is exponentially distributed. This relates \(\lambda\) to \(p\), so the former can be estimated once the latter has been.]

The probability that any particular component of a certain type works in a satisfactory manner is \(p\). If \(n\) of these components are independently selected, then the statistic \(X\), the number among the selected components that perform in a satisfactory manner, is sufficient for \(p\). You must purchase two of these components for a particular system. Obtain an unbiased statistic for the probability that exactly one of your purchased components will perform in a satisfactory manner. [Hint: Start with the statistic \(U\), the indicator function of the event that exactly one of the first two components in the sample of size \(n\) performs as desired, and improve on it by conditioning on the sufficient statistic.]

Six Pepperidge Farm bagels were weighed, yielding the following data (grams): \(\begin{array}{llllll}117.6 & 109.5 & 111.6 & 109.2 & 119.1 & 110.8\end{array}\) a. Assuming that the six bagels are a random sample and the weight is normally distributed, estimate the true average weight and standard deviation of the weight using maximum likelihood. b. Again assuming a normal distribution, estimate the weight below which \(95 \%\) of all bagels will have their weights. [Hint: What is the 95 th percentile in terms of \(\mu\) and \(\sigma\) ? Now use the invariance principle.] c. Suppose we choose another bagel and weigh it. Let \(X=\) weight of the bagel. Use the given data to obtain the mle of \(P(X \leq 113.4)\). (Hint: \(P(X \leq 113.4)=\Phi[(113.4-\mu) / \sigma)] .)\)

Assume that the number of defects in a car has a Poisson distribution with parameter \(\lambda\). To estimate \(\lambda\) we obtain the random sample \(X_{1}\), \(X_{2}, \ldots, X_{n}\). a. Find the Fisher information in a single observation using two methods. b. Find the Cram茅r-Rao lower bound for the variance of an unbiased estimator of \(\lambda\). c. Use the score function to find the mle of \(\lambda\) and show that the mle is an efficient estimator. d. Is the asymptotic distribution of the mle in accord with the second theorem? Explain.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.