The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 84.4 for a sample of size 762 and standard deviation 9.6.
Estimate how much the drug will lower a typical patient's systolic blood pressure (using a $90 \%$ confidence level).
Enter your answer as a tri-linear inequality accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).
Answer
Final Answer: The drug will lower a typical patient's systolic blood pressure by an estimated amount between 83.8 and 85.0 (to one decimal place) with 90% confidence. So, the answer is \(\boxed{83.8 \leq \mu \leq 85.0}\).
Step 1 ：Given that the sample mean (\(\bar{x}\)) is 84.4, the sample standard deviation (s) is 9.6, and the sample size (n) is 762. The z-score for a 90% confidence level is approximately 1.645.
Step 2 ：The formula for the confidence interval for a mean is \(\bar{x} \pm z \frac{s}{\sqrt{n}}\).
Step 3 ：Substitute the given values into the formula to calculate the confidence interval: \(84.4 \pm 1.645 \frac{9.6}{\sqrt{762}}\).
Step 4 ：Calculate the margin of error: 1.645 * (9.6 / \(\sqrt{762}\)) = 0.5720837998193516.
Step 5 ：Calculate the lower bound of the confidence interval: 84.4 - 0.5720837998193516 = 83.82791620018065.
Step 6 ：Calculate the upper bound of the confidence interval: 84.4 + 0.5720837998193516 = 84.97208379981936.
Step 7 ：Round the lower and upper bounds to one decimal place to match the precision of the sample statistics: lower bound = 83.8, upper bound = 85.0.
Step 8 ：Final Answer: The drug will lower a typical patient's systolic blood pressure by an estimated amount between 83.8 and 85.0 (to one decimal place) with 90% confidence. So, the answer is \(\boxed{83.8 \leq \mu \leq 85.0}\).
