a random sample from the population of registered voters in california is to be taken and then surveyed about an upcoming election. What sample size should be used to guarantee a sampling error of 3% or less when estimating p at the 95% confidence level?
is this the answer...196*0.5/.003^2=1067.1
n = (za/2) ^2 * .25/ E^2
n = (1.96)^2 * .25/.03^2
n = .9604/.0009
n = 1068
To calculate the required sample size for a given level of sampling error, confidence level, and population size, you can use the formula:
Sample Size = (Z^2 * p * (1-p)) / (E^2)
Where:
- Z is the z-score corresponding to the desired confidence level
- p is the estimated proportion of the population with the desired attribute
- E is the desired sampling error
In this case, since you want to estimate the proportion of registered voters in California, we don't have an initial estimate for p. Therefore, we use a conservative estimate of p = 0.5 (assuming that 50% of registered voters will support the upcoming election).
Now let's calculate the sample size:
Z-score for a 95% confidence level: Z = 1.96 (based on standard normal distribution tables)
Desired sampling error: E = 0.03 (3% in decimal form)
Sample Size = (1.96^2 * 0.5 * (1-0.5)) / (0.03^2)
Sample Size = 3.8416 * 0.25 / 0.0009
Sample Size = 0.9604 / 0.0009
Sample Size ≈ 1067.11
So, the calculated sample size required to guarantee a sampling error of 3% or less when estimating p at the 95% confidence level is approximately 1067.11.