Question

discrete random variables (10 questions)
the accounting department measures how quarterly returns change when using two different approaches: internal controls (like compliance checks) and automation tools (like ai software). the company simulates four business conditions: (1) audit pressure: during periods of intense audits, (2) steady state: under normal, stable operations, (3) growth phase: when the company is rapidly expanding, (4) compliance boost: when new regulations are implemented. the chart below shows the results.
accounting models quarterly efficiency change (%)
internal controls vs automation tools across four environments
the following 10 questions refer to the data in this chart (or data that can be derived from this chart)
question 2 (1 point)
if the sample size is 9 days, how likely is the sample average to fall between $480 and $520?
about 69%
about 31%
about 33%
about 38%

Explanation:

Step1: Assume normal - distribution (Central Limit Theorem)

For a sample size $n = 9$, if the population has mean $\mu$ and standard - deviation $\sigma$, the sampling distribution of the sample mean $\bar{X}$ has mean $\mu_{\bar{X}}=\mu$ and standard deviation $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}$. According to the empirical rule (68 - 95 - 99.7 rule) for a normal distribution, approximately 68% of the data lies within 1 standard deviation of the mean, i.e., $P(\mu - \sigma_{\bar{X}}<\bar{X}<\mu+\sigma_{\bar{X}})\approx0.68$.
The interval from $480$ to $520$ is symmetric around the mean (assuming the mean is $\frac{480 + 520}{2}=500$). This interval represents an interval that is approximately 1 standard deviation from the mean for the sampling distribution of the sample mean when the sample size $n = 9$.

Answer:

About 69%