Chapter 9A, Problem 15ITC

a.

To determine

Calculate the required sample size using mean-per-unit sampling.

Answer to Problem 15ITC

The required sample size is 230 items using mean-per unit sampling.

Explanation of Solution

Sample size:

The sample size is the portion of the total population that represents the total population. The auditor conducts sampling test on the selected sample. Large sample size would represent the population more accurately but considering a large sample size would be a complex and expensive process. The auditor should select the sample size that would represent the entire population precisely and accurately.

Mean-per-unit sampling:

It is a technique of classical variables sampling. Mean-per-unit sampling works on the assumption that the audited mean of the sample size will represent the actual audited mean of the population. First, the mean of the sample size is calculated then the estimation for the value of the population is calculated by multiplying the calculated sample mean to the total of the items in the population. The difference between the value of the population calculated under mean-per unit sampling and the book value of the client is the projected misstatement.

Calculate the sample size for the population of Company J:

$\begin{array}{c}\text{Sample size = }{\left[\frac{\begin{array}{l}\text{Population size }\times \text{ (Incorrect acceptance coefficient}\\ \text{ + Incorrect rejected coefficient) }\times \text{ Estimated standard deviation}\end{array}}{\text{Tolerable misstatement}}\right]}^2\\ ={\left[\frac{\text{40,000 }\times \text{ (1}\text{.64 + 1}\text{.64) }\times \text{ \$45}}{\$390,000}\right]}^2\\ =\text{ 229}\text{.17 items}\end{array}$

Thus, the required sample size is 230 items using mean-per unit sampling.

b.

To determine

Calculate the required sample size for the given variables.

Answer to Problem 15ITC

The required level of sample size is 93 units.

Explanation of Solution

Calculate the required sample size for the given variables:

The required sample size can be calculated with the help of table provided in the figure 9.4 of the chapter.

Statistical Sample Sizes for Tests of Controls at 5 Percent Risk of Assessing Control Risk Too Low (with Allowable Number of Deviations in Parentheses) shows the required sample size of 93 units at the tolerable deviation rate of 5 percent and risk of assessing control risk at 5%.

Thus, the required level of sample size is 93 units.

c.

To determine

Calculate projected misstatement using mean-per-unit estimation.

Answer to Problem 15ITC

The projected misstatement is $1,057,600.

Explanation of Solution

Mean-per-unit sampling:

It is a technique of classical variables sampling. Mean-per-unit sampling works on the assumption that the audited mean of the sample size will represent the actual audited mean of the population. First, the mean of the sample size is calculated then the estimation for the value of the population is calculated by multiplying the calculated sample mean to the total of the items in the population. The difference between the value of the population calculated under mean-per unit sampling and the book value of the client is the projected misstatement.

Calculate the projected misstatement using mean-per-unit method:

$\begin{array}{c}\text{Projected misstatement = Book value - Audited value}\\ \text{= \$5,000,000 - \$3,942,400}\\ \text{= \$1,057,600}\end{array}$

Thus, the projected misstatement is $1,057,600.

Working note 1:

Calculate the accounts receivable estimated audited value:

$\begin{array}{c}\text{Accounts receivable audited value = Average audited value }\times \text{ Population size}\\ =\frac{\text{Book value - Audited value}}{\text{Sample size}}\times \text{ Population size}\\ \text{= }\frac{\$27,500\text{ - \$7,200}}{208}\times \text{ 40,000}\\ \text{= \$3,942,400}\end{array}$

d.

To determine

Calculate allowance for sampling risk using mean-per-unit estimation.

Answer to Problem 15ITC

The allowable sampling risk is $139,830.

Explanation of Solution

Sampling risk:

It refers to the risk which arises when there are chances of inappropriate sample size. When the sample of the population does not represent the population then the risk of the incorrect conclusion is considered as sampling risk.

Calculate allowance for sampling risk using mean-per-unit estimation:

$\begin{array}{c}\begin{array}{l}\text{Allowable}\\ \text{sampling risk}\end{array})\text{ =}\left(\text{Total misstatement}-\frac{\left(\begin{array}{l}\text{Population size }\times \text{ Incorrect acceptance }\\ \text{coefficient }\times \text{ Sample standard deviation}\end{array}\right)}{\sqrt{\text{Sample size}}}\right)\\ =\text{ \$390,000 - }\frac{\left(40,000\times \text{ 1}\text{.64 }\times \text{ \$55}\right)}{\sqrt{208}}\\ =\text{ \$139,830}\end{array}$

Thus, the allowable sampling risk is $139,830.

e.

To determine

Calculate the projected misstatement using difference estimation.

Answer to Problem 15ITC

The projected misstatement is $3,903,600.

Explanation of Solution

Difference estimation:

Difference estimation is a sampling plan. It is helpful in finding the misstatement in the financial statements of the company. Indifference estimation, the auditor estimates the average difference between the audited value and the book value. The estimated projected misstatement is calculated by multiplying the total number of population to the estimated average difference.

Explain the conclusion that the auditor may draw from using the difference estimation:

Calculate the projected misstatement using difference estimation:

$\begin{array}{c}\text{Projected misstatement = Average difference }\times \text{ Population size}\\ \text{ = }\left(\begin{array}{l}\text{Average book value - Average audited value}\\ \times \text{ Population size}\end{array}\right)\\ \text{= }\left(\left[\frac{\$40,800}{208}\right]\text{ - \$98}\text{.56}\right)\times \text{ 40,000}\\ \text{= \$3,903,600}\end{array}$

Thus, the projected misstatement is $3,903,600.

f.

To determine

Explain the statistical conclusion made by using attributes sampling.

Explanation of Solution

Attributes sampling:

Attributes sampling helps the auditor to assess a specific rate of occurrence for a specific characteristic of the audit population. It is helpful in finding the estimate related to a particular transaction or activity in the system.

Explain the statistical conclusion made by using attributes sampling:

The required sample size can be calculated with the help of table provided in the figure 9.4 of the chapter.

Statistical Sample Sizes for Tests of Controls at 5 Percent Risk of Assessing Control Risk Too Low (with Allowable Number of Deviations in Parentheses) shows the required sample size of 208 units at the tolerable deviation rate of 5 percent and 5 projected misstatement.

Figure 9.5 shows the Statistical Sampling Results Evaluation Table for Tests of Controls: Achieved Upper Deviation Rate at 5 Percent Risk of Assessing Control Risk Too Low table. This table can be used to calculate the upper limit at the deviation of 5 and the sample size of 200 units. It will be 5.2. It states that the deviation can go up to 5.2%.

Thus, the upper deviation rate is 5.2 which is higher than the tolerable misstatement of 5% so the population should not be accepted.

g.

To determine

Calculate projected misstatement, basic precision, incremental allowance and upper limit on misstatement in case when the auditor has discovered the given misstatement.

Answer to Problem 15ITC

1. (1) The projected misstatement is $49,377.
2. (2) The basic precision is $72,114.
3. (3) The incremental allowance is $18,954.
4. (4) The upper limit on misstatement is $140,445.

Explanation of Solution

Reliability factor:

It is used to calculate the sample size. It provides level of confidence at the time of sample results. The numbers of items reviewed are directly correlated to the reliability level.

Basic precision:

It is calculated by multiplying reliability factor to sampling factor. It is a upper level of misstatement when there is zero misstatement in the sample size.

Incremental allowances:

It is calculated by ranking the projected misstatement value with the book value from largest to smallest misstatement.

Calculate projected misstatement, basic precision, incremental allowance and upper limit on misstatement in case when the auditor has discovered the given misstatement:

(1)

Calculate projected misstatement:

Book value (a)	Audited value (b)	Misstatement $\text{c = (a-b)}$	Tainting % $\left(\text{d = }\frac{\text{c}}{\text{a}}\right)$	Sampling interval (e)	Projected misstatement $\text{f = (e }\times \text{ d)}$
$26,000	$5,950	$20,050	-	-	$20,050
$1,000	$900	$100	10%	$24,038	$2,404
$200	$200	$20	9%	$24,038	$2,163
$180	$120	$60	33%	$24,038	$7,933
$100	$30	$70	70%	$24,038	$16,827
					$49,377

Table: (1)

Thus, the projected misstatement is $49,377.

(2)

Calculate the basic precision:

$\begin{array}{c}\text{Basic precision = Reliability factor }\times \text{ Sampling interval}\\ \text{= 3 }\times \text{ \$24,038}\\ \text{= \$72,114}\end{array}$

Thus, the basic precision is $72,114.

(3)

Calculate the incremental allowance:

Reliability factor (a)	Increment $\text{(b = a1 - a2)}$	Increment-1 $\text{(c = 1 - b)}$	Projected misstatement (d)	Incremental allowance $(\text{e = d }\times \text{ c)}$
3
4.75	1.75	0.75	$16,827	$12,620
6.3	1.55	0.55	$7,933	$4,363
7.76	1.46	0.46	$2,404	$1,106
9.16	1.40	0.40	$2,163	$865
				$18,954

Table: (2)

Thus, the incremental allowance is $18,954.

(4)

Calculate the upper limit on misstatement:

$\begin{array}{c}\text{Upper limit on misstatement =}\left\{\begin{array}{c}\text{ Projected misstatement + Basic precision }\\ \text{+ Incremental allowance}\end{array}\right\}\\ =\$49,377+\$72,114+\$18,954\\ =\text{ \$140,445}\end{array}$

Thus, the upper limit on misstatement is $140,445.