Final answer:
To calculate the sample size needed for a 95% confidence interval within 15 mg of the true mean caffeine intake with a standard deviation of 68 mg, use the formula n = (z*σ/E)^2. Approximately 1096 high school students should be sampled.
Step-by-step explanation:
To estimate caffeine consumption in high school students and determine the sample size required to ensure that a 95% confidence interval estimate for the mean caffeine intake is within 15 mg of the true mean, one can use the sample size formula for a mean:
n = (z*σ/E)^2
Where:
- n is the sample size
- z is the z-score corresponding to the desired confidence level (1.96 for 95%)
- σ (sigma) is the population standard deviation
- E is the desired margin of error
Given a standard deviation (σ) of 68 mg and a margin of error (E) of 15 mg:
n = (1.96*68/15)^2
n ≈ 33.08^2
n ≈ 1095.3864
Thus, the investigator would need to sample approximately 1096 students (since we cannot have a fraction of a student) to be 95% confident that the sample mean is within 15 mg of the true population mean caffeine intake.