a. What sample size would be needed to estimate the true proportion of students at your college who are wearing backpacks, with 95% confidence and and error of +/-0.04?
b. What sampling method would you recommend? Why?
a. To determine the sample size needed to estimate the true proportion of students wearing backpacks at your college with 95% confidence and an error of +/-0.04, you would need to use the formula for sample size calculation for proportions.
The formula is given as:
n = (Z^2 * p * q) / E^2
Where:
- n represents the required sample size
- Z is the z-score corresponding to the desired confidence level (95% confidence corresponds to a z-score of approximately 1.96)
- p is the estimated proportion of students wearing backpacks (if unknown, you can assume 0.5 to get the maximum sample size requirement)
- q is 1 - p
- E is the desired margin of error (in this case, +/-0.04)
Let's assume the estimated proportion of students wearing backpacks is 0.5. Plugging in the values, we get:
n = (1.96^2 * 0.5 * 0.5) / (0.04^2)
n ≈ 600.25
Rounding up to the nearest whole number, the required sample size would be approximately 601 students.
b. The sampling method you would recommend depends on the nature of your college and the availability of the student population. Assuming your college has a large number of students, a simple random sampling method would be a good choice.
Simple random sampling ensures that each student in the population has an equal chance of being selected for the sample. This method helps minimize bias and provides a representative sample. If your college has a smaller population, you could consider using stratified sampling, where you divide the population into relevant strata (e.g., different majors, different years) and randomly sample within each stratum.
Ultimately, the choice of sampling method depends on factors such as the college's characteristics, feasibility, and resources available for the research study.