4. Assume you are just planning this survey and want to know how many individuals you have to randomly select to be part of the sample. Assume that the purpose of the study is to calculate a 95% confidence interval for proportion of the adult population who favor lowering the drinking age to 18, such that the margin of error is only 0.03 (or 3% ). We don’t have any idea about what the value of p (the proportion in the population) might be. Calculate the necessary sample size.
To calculate the necessary sample size for a 95% confidence interval with a margin of error of 0.03, we need to make some assumptions and use a formula. Here's how you can calculate it:
1. Determine the level of confidence: In this case, the level of confidence is given as 95%.
2. Determine the margin of error: The margin of error is given as 0.03 (or 3%).
3. Calculate the critical z-value: The critical z-value corresponds to the level of confidence. Since the confidence level is 95%, we need to find the z-value that corresponds to a 95% confidence level. This critical z-value can be found using a z-table or calculator and is typically 1.96 for a 95% confidence level.
4. Calculate the sample size formula: The formula to calculate the necessary sample size is given by:
n = (z^2 * p * (1-p)) / (E^2)
Where:
n = required sample size
z = critical z-value
p = estimated proportion (since we do not have any idea about what the value of p might be, we can use a conservative estimate of 0.5)
E = margin of error
Plugging in the values, we get:
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
So, the necessary sample size is approximately 1068 individuals.