Final answer:
To ensure the Central Limit Theorem applies, an auditor should randomly sample at least 30 customers, as this is the common threshold to assume a normal distribution for the sample mean.
Step-by-step explanation:
A student is curious about how many customers an auditor must sample for the Central Limit Theorem (CLT) to apply when assessing unpaid balances of a credit card agency's customers. The unpaid balance has a bell-shaped distribution with a mean of $5500 and a standard deviation of $500. According to the CLT, the distribution of sample means will tend to be normal, or bell-shaped, if the sample size is sufficiently large, regardless of the shape of the population distribution.
However, the precise sample size needed can depend on various factors. The rule of thumb for the CLT to hold is that the sample size should be large enough. Though there is no specific number universally agreed upon, generally a sample size of 30 or more is considered sufficient for the CLT to apply. Hence, in this scenario, an auditor should sample at least 30 customers to ensure that the CLT will apply.
The importance of the CLT lies in its ability to justify the use of normal probability distributions for inference about the population mean when dealing with sufficiently large sample sizes. This is beneficial because it allows for easier calculation of probabilities and the application of confidence intervals and hypothesis tests.