Ethan needs to conduct a study to
determine whether the students in his school like playing football. He decides to divide the students
into two groups, boys and girls, knowing that the school has a total of 200
students, 80 of whom are girls. If Ethan decides that his sample
size will be 50, how many girls should he select for the study?
Since the population of students is
split into two distinct strata, that is, boys and girls, the appropriate sampling
method is stratified or layered random sampling. A stratified random sample is a
sample consisting of random samples selected from distinct groups or strata within
the population. The sample size for each stratum
reflects the stratum proportion of the population.
In order to calculate the sample
size for a particular stratum, that is, lowercase 𝑠, we use the formula lowercase
𝑠 is equal to uppercase 𝑆, which is the number in the stratum, divided by the
number in the population, uppercase 𝑁, multiplied by lowercase 𝑛, which is the
overall sample size. In our case, we have a total of 200
students so that uppercase 𝑁 is 200. We know that we have 80 girls so
that uppercase 𝑆 is equal to 80 and that Ethan’s sample size is 50. That is, lowercase 𝑛 is equal to
50. And our sample size for girls is 80
over 200 multiplied by 50, that is, 80 girls divided by a population of 200
multiplied by the sample size 50. We can divide numerator and
denominator by 50 and again numerator and denominator by four, which gives us
20. Therefore, Ethan should select 20
girls for his study.