A developer claims that the percent of city residents who favor building a new football stadium is between 52.3 % and 61.7 %. We need to find the number of residents surveyed. First, recall that when a random sample of size n is taken from a large population, the margin of error can be approximated with the following formula.
Margin of error = ± 1/sqrt(n)
Now, this means that if the percent of the sample responding a certain way is p, then the percent of the population who would respond the same way is likely to be less than the margin of error away from p. Therefore, it is likely to be between the two following values.
p- 1/sqrt(n) and p+ 1/sqrt(n)Notice that we already know both of the values, since we were told that the percent of city residents who favor building a new football stadium is between 52.3 % and 61.7 %.
p+ 1/sqrt(n) &= 61.7 %
p- 1/sqrt(n) &= 52.3 %
Let's now subtract the two equalities and then simplify. This will being us closer to finding the value of n.
