A community activist group in Austin, Texas wanted a particular issue to be placed on the ballot of the upcoming election. To make it on the ballot, 20,000 valid signatures were needed.The group turned in their petition with 24,598 signatures. To pass the validity test 20,000/24,598 = 81.3% of the signatures must be valid. It is too time consuming to check all of the signatures, so a random sample of signatures are checked. The individual checking the signatures needs to be 95% confident that the true proportion of valid signatures are estimated with, at most, a 2% margin of error.

1. Using a conservative estimate for p-hat , how large of a sample is needed?
2. In the activist group's previous petition, 85% of the signatures were valid. Using this value as a guess for p-hat , find the sample size needed for a margin of error of at most 2 percentage points with 95% confidence. How does this compare with the required sample size from Question 1?