Below we have the results of a survey, in which people were asked whether they prefer sports drink X or sports drink Y. We want to know how big the sample needs to be in order for us to be sure that sports drink X is truly preferred by more than half of the population.
Sports Drink X
Sports Drink Y
52 %
48 %
First, recall that when a 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, if the percent of the people choosing sports drink X over sports drink Y is 52 %, then the percent of the population who would respond the same way is likely to be less than the margin of error away from 52 %. Therefore, it is likely to be between the two following values.
52 %- 1/sqrt(n) and 52 %+ 1/sqrt(n)
In the best scenario, 52 %+ 1sqrt(n) of the population prefers sports drink X. In the worst scenario, only 52 %- 1sqrt(n) prefers the drink. Therefore, to be confident that sports drink X is truly preferred, the lower margin would need to be higher than 50 %.
52 %- 1/sqrt(n) > 50 %
Let's solve the inequality.
We found that at least 2500 people need to participate in the survey in order for us to be confident that sports drink X is truly preferred by more than half of the population.
