Chapter G, Problem 50EP

a.

To determine

Calculate the sample size and sampling interval.

Explanation of Solution

The sample size: The sample size signifies the number of items that the audit team inspects within a population of interest. Four major aspects that impact the sample size in an “attributes sampling application” are as follows:

• “Tolerable rate of deviation”.
• “Sampling risk-risk of overreliance, or risk of assessing control risk too low.
• “Expected population deviation rate”.
• “Population size”.

Sampling interval: The samples units from a larger population are picked corresponding to an arbitrary beginning point, however, with a fixed interval. This fixed interval is referred to as sampling interval.

• Risk of incorrect acceptance = 5 percent.

Calculate the ratio of expected to tolerable misstatement.

$\begin{array}{c}\left(\begin{array}{l}\text{Ratio of Expected to}\\ \text{Tolerable Misstatement}\end{array}\right)=\frac{\text{Expected misstatement}}{\text{Tolerable misstatement}}\\ =\frac{\$50,000}{\$250,000}\\ =0.20\end{array}$

Calculate the Tolerable misstatement as a percentage of population.

$\begin{array}{c}\left(\begin{array}{l}\text{Tolerable Misstatement As }\\ \text{a Percentage of Population}\end{array}\right)=\frac{\text{Tolerable misstatement}}{\text{Accounts receivable}}\\ =\frac{\$250,000}{10,000,000}\\ =2.5\%\end{array}$

Sample size ranges from 155 (tolerable misstatement 3% of population) to 232 (tolerable misstatement 2% of population) to interpolate.

Calculate the sample size.

Step 1: Calculate the tolerable misstatement 1%.

$\begin{array}{c}\text{Tolerable Misstatement of 1\%}=3\%-2\%\\ =0.03-0.02\\ =0.01\end{array}$

Step 2: Calculate the tolerable misstatement 5%.

$\begin{array}{c}\text{Tolerable Misstatement of 5\%}=\frac{\left(2.5\%-2\%\right)}{\text{Tolerable Misstatement 1\%}}\\ =\frac{0.025-0.02}{0.01}\\ =0.50\end{array}$

Step 3: Calculate the tolerable misstatement.

$\begin{array}{c}\text{Tolerable Misstatement}=\left(\begin{array}{l}\text{Sample size range of 2\%}\\ -\text{Sample size range of 3\%}\end{array}\right)\times \left(\begin{array}{l}\text{Tolerable }\\ \text{Misstatement of 5\%}\end{array}\right)\\ =\left(232-155\right)\times 0.50\\ =38.5\end{array}$

Step 4: Calculate the sample size.

$\begin{array}{c}\text{Sample Size}=\left(\begin{array}{l}\text{Sample size range 2\% }\\ -\text{Tolerable misstatement}\end{array}\right)\\ =232-38.5\\ =194\end{array}$

Step 5: Calculate the sampling interval.

$\begin{array}{c}\text{Sampling Interval}=\frac{\text{Recorded balance}}{\text{Sample size}}\\ =\frac{\$10,000,000}{194}\\ =\$51,546\left(\text{Rounded}\right)\end{array}$

Note: In case the student doesn’t interpolate and uses the conservative high sample size of 232, the sampling interval will be $43,103 (working note: 1).

Working Note 1: Calculate the sampling interval.

$\begin{array}{c}\text{Sampling Interval}=\frac{\text{Recorded balance}}{\text{Sample size}}\\ =\frac{\$10,000,000}{232}\\ =\$43,103\left(\text{Rounded}\right)\end{array}$

b.

To determine

Explain how Mr. D will select customer accounts from the population of accounts receivable balances for confirmation based on the calculation in requirement (a).

Explanation of Solution

First person D will choose a random starting point among the population using either a random number table or computerized random number generator. Mr. D’s first selection is the logical unit containing this dollar. Mr. D makes an additional selection with the first item by skipping $51,546 in the population and next selection would be logical unit which contains this dollar. Until 194 items are selected this process will repeat.

c1.

To determine

Determine the sample size and sampling interval, if risk of incorrect acceptance is 10%.

Explanation of Solution

• Risk of incorrect acceptance = 10 percent.

Calculate the ratio of expected to tolerable misstatement.

$\begin{array}{c}\left(\begin{array}{l}\text{Ratio of Expected to}\\ \text{Tolerable Misstatement}\end{array}\right)=\frac{\text{Expected misstatement}}{\text{Tolerable misstatement}}\\ =\frac{\$50,000}{\$250,000}\\ =0.20\end{array}$

Calculate the Tolerable misstatement as a percentage of population.

$\begin{array}{c}\left(\begin{array}{l}\text{Tolerable Misstatement As }\\ \text{a Percentage of Population}\end{array}\right)=\frac{\text{Tolerable misstatement}}{\text{Accounts receivable}}\\ =\frac{\$250,000}{10,000,000}\\ =2.5\%\end{array}$

Sample size ranges from 114 (tolerable misstatement 3% of population) to 171 (tolerable misstatement 2% of population) to interpolate.

Step 1: Calculate the tolerable misstatement 1%.

$\begin{array}{c}\text{Tolerable Misstatement of 1\%}=3\%-2\%\\ =0.03-0.02\\ =0.01\end{array}$

Step 2: Calculate the tolerable misstatement 5%.

$\begin{array}{c}\text{Tolerable Misstatement of 5\%}=\frac{\left(2.5\%-2\%\right)}{\text{Tolerable Misstatement 1\%}}\\ =\frac{0.025-0.02}{0.01}\\ =0.50\end{array}$

Step 3: Calculate the tolerable misstatement.

$\begin{array}{c}\text{Tolerable Misstatement}=\left(\begin{array}{l}\text{Sample size range of 2\%}\\ -\text{Sample size range of 3\%}\end{array}\right)\times \left(\begin{array}{l}\text{Tolerable }\\ \text{Misstatement of 5\%}\end{array}\right)\\ =\left(171-114\right)\times 0.50\\ =28.5\end{array}$

Step 4: Calculate the sample size.

$\begin{array}{c}\text{Sample Size}=\left(\begin{array}{l}\text{Sample size range 2\% }\\ -\text{Tolerable misstatement}\end{array}\right)\\ =171-28.5\\ =143\end{array}$

Step 5: Calculate the sampling interval.

$\begin{array}{c}\text{Sampling Interval}=\frac{\text{Recorded balance}}{\text{Sample size}}\\ =\frac{\$10,000,000}{143}\\ =\$69,930\left(\text{Rounded}\right)\end{array}$

Note: In case the student doesn’t interpolate and uses the conservative high sample size of 171, the sampling interval will be $58,479 (working note: 2).

Working Note 2: Calculate the sampling interval.

$\begin{array}{c}\text{Sampling Interval}=\frac{\text{Recorded balance}}{\text{Sample size}}\\ =\frac{\$10,000,000}{171}\\ =\$58,479\left(\text{Rounded}\right)\end{array}$

c2.

To determine

Determine the sample size and sampling interval, if the tolerable misstatement decreases from $250,000 to $125,000.

Explanation of Solution

• Risk of incorrect acceptance = 5 percent.

Calculate the ratio of expected to tolerable misstatement.

$\begin{array}{c}\left(\begin{array}{l}\text{Ratio of Expected to}\\ \text{Tolerable Misstatement}\end{array}\right)=\frac{\text{Expected misstatement}}{\text{Tolerable misstatement}}\\ =\frac{\$50,000}{\$125,000}\\ =0.40\end{array}$

Calculate the Tolerable misstatement as a percentage of population.

$\begin{array}{c}\left(\begin{array}{l}\text{Tolerable Misstatement As }\\ \text{a Percentage of Population}\end{array}\right)=\frac{\text{Tolerable misstatement}}{\text{Accounts receivable}}\\ =\frac{\$125,000}{10,000,000}\\ =1.25\%\end{array}$

Sample size ranges from 405 (tolerable misstatement 2% of population) to 809 (tolerable misstatement 1% of population) to interpolate.

Step 1: Calculate the tolerable misstatement 1%.

$\begin{array}{c}\text{Tolerable Misstatement of 1\%}=2\%-1\%\\ =0.02-0.01\\ =0.01\end{array}$

Step 2: Calculate the tolerable misstatement 25%.

$\begin{array}{c}\text{Tolerable Misstatement of 5\%}=\frac{\left(1.25\%-1\%\right)}{\text{Tolerable Misstatement 1\%}}\\ =\frac{0.0125-0.01}{0.01}\\ =0.25\end{array}$

Step 3: Calculate the tolerable misstatement.

$\begin{array}{c}\text{Tolerable Misstatement}=\left(\begin{array}{l}\text{Sample size range of 1\%}\\ -\text{Sample size range of 2\%}\end{array}\right)\times \left(\begin{array}{l}\text{Tolerable }\\ \text{Misstatement of 25\%}\end{array}\right)\\ =\left(809-101\right)\times 0.25\\ =101\end{array}$

Step 4: Calculate the sample size.

$\begin{array}{c}\text{Sample Size}=\left(\begin{array}{l}\text{Sample size range 1\% }\\ -\text{Tolerable misstatement}\end{array}\right)\\ =809-101\\ =708\end{array}$

Step 5: Calculate the sampling interval.

$\begin{array}{c}\text{Sampling Interval}=\frac{\text{Recorded balance}}{\text{Sample size}}\\ =\frac{\$10,000,000}{708}\\ =\$14,124\left(\text{Rounded}\right)\end{array}$

Note: In case the student doesn’t interpolate and uses the conservative high sample size of 809, the sampling interval will be $12,360 (working note: 3).

Working Note 3: Calculate the sampling interval.

$\begin{array}{c}\text{Sampling Interval}=\frac{\text{Recorded balance}}{\text{Sample size}}\\ =\frac{\$10,000,000}{809}\\ =\$12,360\left(\text{Rounded}\right)\end{array}$

c3.

To determine

Determine the sample size and sampling interval, if the expected misstatement decreases from $50,000 to $25,000.

Explanation of Solution

• Risk of incorrect acceptance = 5 percent.

Calculate the ratio of expected to tolerable misstatement.

$\begin{array}{c}\left(\begin{array}{l}\text{Ratio of Expected to}\\ \text{Tolerable Misstatement}\end{array}\right)=\frac{\text{Expected misstatement}}{\text{Tolerable misstatement}}\\ =\frac{\$25,000}{\$250,000}\\ =0.410\end{array}$

Calculate the Tolerable misstatement as a percentage of population.

$\begin{array}{c}\left(\begin{array}{l}\text{Tolerable Misstatement As }\\ \text{a Percentage of Population}\end{array}\right)=\frac{\text{Tolerable misstatement}}{\text{Accounts receivable}}\\ =\frac{\$250,000}{10,000,000}\\ =2.5\%\end{array}$

Sample size ranges from 123 (tolerable misstatement 3% of population) to 184 (tolerable misstatement 2% of population) to interpolate.

$\begin{array}{l}0.03–0.02=0.01\hfill \\ \left(0.025–0.02\right)/0.01=0.50\hfill \\ \left(184–123\right)\times 0.50=30.5\hfill \\ 184–30=154\left(\text{rounded up}\right)\hfill \end{array}$

Step 1: Calculate the tolerable misstatement 1%.

$\begin{array}{c}\text{Tolerable Misstatement of 1\%}=3\%-2\%\\ =0.03-0.02\\ =0.01\end{array}$

Step 2: Calculate the tolerable misstatement 5%.

$\begin{array}{c}\text{Tolerable Misstatement of 5\%}=\frac{\left(2.5\%-2\%\right)}{\text{Tolerable Misstatement 1\%}}\\ =\frac{0.025-0.02}{0.01}\\ =0.50\end{array}$

Step 3: Calculate the tolerable misstatement.

$\begin{array}{c}\text{Tolerable Misstatement}=\left(\begin{array}{l}\text{Sample size range of 2\%}\\ -\text{Sample size range of 3\%}\end{array}\right)\times \left(\begin{array}{l}\text{Tolerable }\\ \text{Misstatement of 5\%}\end{array}\right)\\ =\left(184-123\right)\times 0.50\\ =30.5\end{array}$

Step 4: Calculate the sample size.

$\begin{array}{c}\text{Sample Size}=\left(\begin{array}{l}\text{Sample size range 2\% }\\ -\text{Tolerable misstatement}\end{array}\right)\\ =184-30\\ =154\end{array}$

Step 5: Calculate the sampling interval.

$\begin{array}{c}\text{Sampling Interval}=\frac{\text{Recorded balance}}{\text{Sample size}}\\ =\frac{\$10,000,000}{154}\\ =\$64,935\left(\text{Rounded}\right)\end{array}$

Note: In case the student doesn’t interpolate and uses the conservative high sample size of 184, the sampling interval will be $54,347 (working note: 4).

Working Note 4: Calculate the sampling interval.

$\begin{array}{c}\text{Sampling Interval}=\frac{\text{Recorded balance}}{\text{Sample size}}\\ =\frac{\$10,000,000}{184}\\ =\$54,347\left(\text{rounded}\right)\end{array}$

d.

To determine

Explain how the changes are noted in requirement 3 by describing the relationship between sample size and various factors.

Explanation of Solution

The relationship between sample size and various factors are recognized by relating the sample size in (c) and with the calculation in (a), are as follows:

• The sample size decreases from 194 to 143(refer [c1]). The risk of incorrect acceptance will have an inverse relationship with sample size when the acceptable risk of incorrect acceptance increases from 5 percent to 10 percent.
• The sample size increases from 194 to 708 (refer [c2]). The tolerable misstatement will have an inverse relationship with sample size when the tolerable misstatement decreases from $250,000 to $125,000.
• The sample size decreases from 194 to 154 (refer [c3]). The expected misstatement will have an inverse relationship with sample size when the tolerable misstatement decreases from $50,000 to $25,000.

e.

To determine

Explain the relationship between sample size and sampling interval. Briefly explain as to the nature of this relationship.

Explanation of Solution

There is an inverse relationship between the sample size and sampling interval. This relationship is observed because a large number of samples must be taken from the sample population, in order to select the additional items a smaller number of items have to be skipped.