Final answer:
The point estimate of the mean time for caffeine to leave the body is 6.0 hours. To calculate the sample size that provided the 99% confidence interval estimate with a population standard deviation of 2 hours, the formula involving the z-score and margin of error yields a sample size of approximately 68.
Step-by-step explanation:
A Time magazine report on the mean time for caffeine to leave the body after consumption has provided a 99% confidence interval estimate of the population mean time between 5.6 and 6.4 hours. To answer the student's questions:
a. Point Estimate of the Mean Time
The point estimate of the mean time for caffeine to leave the body is the midpoint of the confidence interval. This is calculated by averaging the lower and upper limits of the interval; in this case, (5.6 hours + 6.4 hours) / 2, which equals 6.0 hours.
b. Sample Size for Interval Estimate
Given the population standard deviation (σ) of 2 hours, and the width of the confidence interval (which is the difference between the upper and lower limits, or 6.4 hours - 5.6 hours = 0.8 hours), we can calculate the sample size (n) used to provide the interval estimate. We use the following formula for sample size:
Where 'z' is the z-score corresponding to the confidence level (for a 99% confidence interval, it is typically 2.576), and E is the margin of error (width of the confidence interval). Solving for n:
- n = ((2.576 * 2) / (0.8/2))^2
- n = approximately 67.6
Therefore, the sample size used to estimate the confidence interval is approximately 68.