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.
Alpha = ________
Z= ________
p= ________
Effect Size = ________
n= ________
alpha = a= 0.05
Z0.025 = 1.96
p = 0.5
q = 1-p = 0.5
E = 0.05
Effect size
= 384.16
n = 385
To answer this question, we need to use the formula to calculate the sample size required for estimating a proportion. The formula is as follows:
n = (Z^2 * p * (1-p)) / (E^2)
Where:
n = sample size
Z = Z-value for the desired level of confidence (in this case, 95% confidence)
p = estimated proportion (we will assume 0.5 as a conservative estimate since we do not have prior information)
E = maximum allowable error (in this case, 5% or 0.05)
Now let's fill in the values:
Alpha = 1 - Confidence Level = 1 - 0.95 = 0.05 (5% significance level)
Z = Z-value for 95% confidence level = 1.96 (from standard normal distribution table)
p = estimated proportion = 0.5 (assuming 50% of high school students consume coffee)
Effect Size = E = maximum allowable error = 0.05 (5%)
Substituting these values into the formula, we have:
n = (1.96^2 * 0.5 * (1-0.5)) / (0.05^2)
Simplifying the calculation:
n = (3.8416 * 0.25) / 0.0025
n = 384.16
Since we cannot have a fractional number of students, we need to round up to the nearest whole number. Therefore, the required sample size would be 385 students.
Summary:
Alpha = 0.05
Z = 1.96
p = 0.5
Effect Size = 0.05
n = 385