2. An investigator wants to estimate caffeine consumption in high school students. How many students would be required to estimate the proportion of students who consume coffee? Suppose we want the estimate to be within 5% of the true proportion with 95% confidence.

To estimate the required sample size for estimating the proportion of high school students who consume coffee, we can use a formula for calculating sample size for proportions.

The formula to calculate the sample size required for estimating a proportion with a desired level of confidence and precision is:

n = (Z^2 * p * (1-p)) / E^2

Where:
n = sample size
Z = Z-value corresponding to the desired level of confidence (in this case, 95% confidence level)
p = estimated proportion (this can be determined based on previous research or assumed to be 0.5 for maximum sample size)
E = maximum error or precision (in this case, 5% or 0.05)

Let's calculate the sample size:

Z = 1.96 (corresponding to a 95% confidence level)
p = 0.5 (maximum sample size assumption)
E = 0.05 (maximum error or precision)

n = (1.96^2 * 0.5 * (1-0.5)) / (0.05^2)
n = (3.8416 * 0.25) / 0.0025
n = 0.9604 / 0.0025
n = 384.16

So, the investigator would need a sample size of approximately 384 students to estimate the proportion of high school students who consume coffee with a 95% confidence level and an error margin of 5%.

Since a sample size must be a whole number, rounding up to the nearest whole number:

n = ceil(384.16) = 385

Therefore, approximately 385 students would be required to estimate the proportion of high school students who consume coffee with a 95% confidence level and an error margin of 5%.

To estimate the required sample size to estimate the proportion of students who consume coffee with a desired level of precision and confidence, we can use the formula for sample size calculation:

n = (Z^2 * p * (1 - p)) / E^2

Where:
n = required sample size
Z = Z-value corresponding to the desired level of confidence (95% confidence corresponds to a Z-value of 1.96)
p = estimated proportion (if no prior estimate is available, 0.5 can be used for maximum variability and maximum required sample size)
E = desired level of precision (margin of error; 5% can be converted to 0.05)

Plugging in the values, we can calculate the required sample size:

n = (1.96^2 * 0.5 * (1 - 0.5)) / 0.05^2

n = (3.8416 * 0.25) / 0.0025

n = 0.9604 / 0.0025

n ≈ 384.16

Therefore, approximately 385 high school students would be required to estimate the proportion of students who consume coffee with a 95% confidence level and a 5% margin of error. Since sample sizes should be integers, rounding up to the nearest whole number, the investigator would need a sample size of at least 385 students.