When doing a sample size estimation for population proportion, what happens when the resultant confidence interval doesn’t contain the sample proportion that was used to come up with the sample size? For example:
For the next poll of the president's approval rating, we want to get a margin of error of 1% with 95% confidence. How many individuals should we sample? In the last poll his approval rate was 72%.
- Confidence level: 95%
- MoE: 1%
- Sample proportion (p-hat): 72%
- Sample size (n): 7,745
We conduct a new poll with sample size of 7,745, and the approval rate is 60%, so the confidence interval is 59 to 61% -- which doesn't contain the sample proportion of 72% that was used to estimate the sample size.
Does this mean the new approval rate of 59 to 61% has to be discarded, and a new poll with a more conservative sample proportion of 60% has to be done?