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 51 such drivers in 2009 and determines the sample mean BAC to be $0.15 \mathrm{~g} / \mathrm{dL}$. with a standard deviation of $0.080 \mathrm{~g} / \mathrm{dL}$. Complete parts (a) through (d) below.
along with the fact that the data were obtained using a simple random sample, satisfies the requirements for constructing a confidence interval.
A. The sample size is likely greater than $10 \%$ of the population.
B. The sample size is likely less than $10 \%$ of the population.
C. The sample size is likely less than $5 \%$ of the population.
D. The sample size is likely greater than $5 \%$ 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 $\square \%$ confident that the population mean BAC is not between $\square$ and $\square$ for drivers involved in fatal accidents who have a positive BAC value.
B. There is a $\square \%$ 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 $\square \%$ confident that the population mean BAC is between $\square$ and $\square$ for drivers involved in fatal accidents who have a positive BAC value.
Answer
\(\boxed{\text{Final Answer: The researcher is 90% confident that the population mean BAC is between 0.132 and 0.169 for drivers involved in fatal accidents who have a positive BAC value.}}\)
Step 1 ：Given that the sample mean BAC is 0.15 g/dL, the standard deviation is 0.08 g/dL, and the sample size is 51 drivers.
Step 2 ：We are asked to determine a 90% confidence interval for the mean BAC in fatal crashes in which the driver had a positive BAC.
Step 3 ：The formula for a confidence interval is given by \( \text{sample mean} \pm z \times \left(\frac{\text{standard deviation}}{\sqrt{\text{sample size}}}\right) \).
Step 4 ：The z-value for a 90% confidence interval is 1.645.
Step 5 ：Substituting the given values into the formula, we get \( 0.15 \pm 1.645 \times \left(\frac{0.08}{\sqrt{51}}\right) \).
Step 6 ：Solving the above expression, we get the confidence interval as \( (0.1315723140941914, 0.1684276859058086) \).
Step 7 ：Rounding to three decimal places, we get the confidence interval as \( (0.132, 0.169) \).
Step 8 ：\(\boxed{\text{Final Answer: The researcher is 90% confident that the population mean BAC is between 0.132 and 0.169 for drivers involved in fatal accidents who have a positive BAC value.}}\)
