Final answer:
The acceptable risk of assessing control risk too low is inversely related to sample size; as ARACR decreases, the required sample size increases to maintain the same level of confidence and power in the statistical test.
Step-by-step explanation:
The relationship of acceptable risk of assessing control risk too low (ARACR) to sample size is such that as the ARACR increases, the sample size needed for a certain level of confidence also increases. When an auditor sets a lower ARACR, they are indicating that they want to reduce the risk of concluding that the controls are more effective than they actually are.
Consequently, to achieve a lower ARACR, auditors will need to increase their sample size. This is because increasing the sample size reduces the margin of error and increases the power of the test, which is the probability of correctly rejecting the null hypothesis when it is false.
Likewise, if auditors desire a smaller error bound while maintaining the same level of confidence, they must increase the sample size. This approach is supported by statistical literature which suggests that, under certain conditions, larger sample sizes can provide more reliable results and better estimates of population parameters.
However, it is important to note that sample size adequacy depends on the specific context of the study including the variability of the data and the desired precision of the results.