Severe acute respiratory syndrome (SARS) is a viral respiratory illness. A random sample of 81 SARS patients was obtained, and researchers found that the mean incubation period was 4.6 days, with a standard deviation of 15.9 days. Based on this information, construct a $95 \%$ confidence interval for the mean incubation period of the SARS virus. Interpret this interval.
Please state the conclusion of the confidence interval in the space below.
Answer
Final Answer: The 95% confidence interval for the mean incubation period of the SARS virus is \(\boxed{[1.14, 8.06]}\) days. This means that we are 95% confident that the true mean incubation period of the SARS virus is between 1.14 and 8.06 days.
Step 1 ：Given that the sample mean (\(\bar{x}\)) is 4.6 days, the standard deviation (\(\sigma\)) is 15.9 days, and the sample size (\(n\)) is 81.
Step 2 ：We are asked to construct a 95% confidence interval for the mean incubation period of the SARS virus. The z-score corresponding to a 95% confidence level is 1.96.
Step 3 ：We can use the formula for a confidence interval for a population mean: \(\bar{x} \pm z \frac{\sigma}{\sqrt{n}}\)
Step 4 ：Substituting the given values into the formula, we get: \(4.6 \pm 1.96 \frac{15.9}{\sqrt{81}}\)
Step 5 ：Solving the above expression, we get the lower and upper bounds of the confidence interval as approximately 1.14 and 8.06 respectively.
Step 6 ：Final Answer: The 95% confidence interval for the mean incubation period of the SARS virus is \(\boxed{[1.14, 8.06]}\) days. This means that we are 95% confident that the true mean incubation period of the SARS virus is between 1.14 and 8.06 days.
