Many ancient tombs were cut from limestone rock that contained uranium. Since most such tombs are not well-ventilated, they may contain radon gas. In one study, the radon levels in a sample of 12 tombs in a particular region were measured in becquerels per cubic meter $\left(\mathrm{Bq} / \mathrm{m}^{3}\right)$. For this data, assume that $\bar{x}=3,560 \mathrm{~Bq} / \mathrm{m}^{3}$ and $\mathrm{s}=1,300 \mathrm{~Bq} / \mathrm{m}^{3}$. Use this information to estimate, with $95 \%$ confidence, the true mean level of radon exposure in tombs in the region. Interpret the resulting interval. Assume that the sampled population is approximately normal.
The confidence interval is $(\square, \square)$.
(Round to the nearest integer as needed.)
Answer
This means that we are 95% confident that the true mean level of radon exposure in tombs in the region lies between 2824 Bq/m³ and 4296 Bq/m³.
Step 1 ：The problem is asking for a 95% confidence interval for the mean level of radon exposure in tombs in the region.
Step 2 ：The confidence interval can be calculated using the formula for a confidence interval for a mean, which is \(\bar{x} \pm z \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(s\) is the sample standard deviation, \(n\) is the sample size, and \(z\) is the z-score corresponding to the desired level of confidence.
Step 3 ：In this case, \(\bar{x} = 3560 \, Bq/m^3\), \(s = 1300 \, Bq/m^3\), \(n = 12\), and the z-score for a 95% confidence interval is approximately 1.96.
Step 4 ：Substituting these values into the formula gives a lower bound of 2824 and an upper bound of 4296 for the confidence interval.
Step 5 ：Final Answer: The 95% confidence interval for the true mean level of radon exposure in tombs in the region is \(\boxed{2824}, \boxed{4296}\) Bq/m³.
Step 6 ：This means that we are 95% confident that the true mean level of radon exposure in tombs in the region lies between 2824 Bq/m³ and 4296 Bq/m³.
