A researcher wishes to estimate the average blood alcohol concentration (BAC) for drivers involved in fatal accidents who are found to have positive BAC values. He randomly selects records from 82 such drivers in 2009 and determines the sample mean BAC to be $0.17 \mathrm{~g} / \mathrm{dL}$ with a standard deviation of $0.060 \mathrm{~g} / \mathrm{dL}$. Complete parts (a) through (d) below.
D. The sample size is likely greater than $10 \%$ of the population.
(c) Determine and interpret a $90 \%$ confidence interval for the mean BAC in fatal crashes in which the driver had a positive BAC. Select the correct choice below and fill in the answer boxes to complete your choice. (Use ascending order. Round to three decimal places as needed.)
A. The researcher is $\%$ confident that the population mean BAC is not between $\square$ and for drivers involved in fatal accidents who have a positive BAC value.
B. There is a $\%$ probability that the population mean BAC is between $\square$ and $\square$ for drivers involved in fatal accidents who have a positive BAC value.
C. The researcher is $\%$ confident that the population mean BAC is between $\square$ and for drivers involved in fatal accidents who have a positive BAC value.
Answer
\(\boxed{\text{Final Answer: The researcher is 90% confident that the population mean BAC is between 0.159 and 0.181 for drivers involved in fatal accidents who have a positive BAC value.}}\)
Step 1 ：Given that the sample mean (\(\bar{x}\)) is 0.17, the standard deviation (s) is 0.06, and the sample size (n) is 82. The z-score for a 90% confidence interval is approximately 1.645.
Step 2 ：We can calculate the margin of error using the formula \(z \frac{s}{\sqrt{n}}\). Substituting the given values, we get a margin of error of approximately 0.0109.
Step 3 ：We can then calculate the lower and upper bounds of the confidence interval by subtracting and adding the margin of error from the sample mean respectively. This gives us a lower bound of approximately 0.159 and an upper bound of approximately 0.181.
Step 4 ：\(\boxed{\text{Final Answer: The researcher is 90% confident that the population mean BAC is between 0.159 and 0.181 for drivers involved in fatal accidents who have a positive BAC value.}}\)
