How large a sample should be surveyed to estimate the true proportion of college students who do laundry once a week within 3% with 95% confidence?
To estimate the true proportion of college students who do laundry once a week with a certain level of confidence and a specific margin of error, you can use the formula for sample size calculation:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = required sample size
Z = Z-score corresponding to the desired level of confidence (in this case, 95% confidence corresponds to a Z-score of approximately 1.96)
p = estimated proportion (since a specific estimate is not given, we can assume p = 0.5, which provides the maximum sample size)
E = margin of error, given as 3% or 0.03
Now we can plug the values into the formula:
n = (1.96^2 * 0.5 * (1-0.5)) / 0.03^2
n = (3.8416 * 0.25) / 0.0009
n = 0.9604 / 0.0009
n ≈ 1067.11
The required sample size is approximately 1068. Therefore, you should survey a sample of around 1068 college students to estimate the true proportion of college students who do laundry once a week within 3% (with 95% confidence).
To determine the sample size needed to estimate the true proportion with a desired level of precision and confidence, we can use the following formula:
n = (Z^2 * p * (1 - p)) / (E^2)
Where:
- n is the required sample size
- Z is the z-value corresponding to the desired level of confidence (95% confidence corresponds to a z-value of 1.96)
- p is the estimated proportion of college students who do laundry once a week (we can use 0.5 as a conservative estimate to maximize sample size)
- E is the desired level of precision (in this case, ±3% or 0.03)
Substituting the values in the formula:
n = (1.96^2 * 0.5 * (1 - 0.5)) / (0.03^2)
n = (3.8416 * 0.5 * 0.5) / 0.0009
n = (0.9604) / 0.0009
n ≈ 1067
Therefore, a sample size of approximately 1067 college students should be surveyed to estimate the true proportion of college students who do laundry once a week within 3% with 95% confidence.