Question

2. Suppose you work for the Economist/YouGov News polling agency gives you the task to conduct this poll again a month later in such a way such that it yields an estimate with a margin of error of at most 1.5% at the 88% confidence level. How many people, at a minimum, should this new poll be conducted on? Before calculating this, predict intuitively if it will be more, less, or the same as 1500 - the number of people in the given poll - and give an explanation. Then calculate the required sample size.

3. Now suppose the agency gives you the task to conduct this poll again a month later in such a way such that it yields an estimate with a margin of error of at most 1.5% at the 99% confidence level. How many people, at a minimum, should this new poll be conducted on? Before calculating this, predict intuitively if it will be more, less, or the same as the number people found in part 2 and give an explanation. Then calculate the required sample size

Answer 1

Q2) First without calculations as the Margin of error here is too low, that is 0.015 or 1.5%, and the sample size is inversely proportional to the square of margin of error, therefore from first look it seems like that the sample size required would be more than 1500.

For 88% confidence level, we have from the standard normal tables:
P(Z < 1.555) = 0.94

Therefore, due to symmetry, we get here:
P( -1.555 < Z < 1.555) = 0.88

The margin of error here is computed as:

For no prior proportion value, we use p = 0.5 to get a conservative value of sample size n here as:

Therefore 2687 is the required sample size here.

Q3) For more confidence level, the critical z value increases and as sample size is directly proportional to the square of critical value, therefore the sample size required would also increase. The new sample size required is computed here as:

From standard normal tables, we have:
P( -2.576 < Z < 2.576) = 0.99

Therefore 7374 is the required sample size here.