In an HR study about the salaries in a certain company, the employees are divided into males and females. The total percentage of females in the company is 60 percent. A sample of 10 employees is selected from the company. The males in that sample represent five percent of the males in the company. What is the total number of employees in that company?
To begin with, we note that 60 percent of the employees in the company are female and that the employees are divided into males and females so that if 60 percent are female, then 100 minus 60 percent, that’s 40 percent of employees, are male. Now, if we let 𝑁 equal the total number of employees in the company and let uppercase 𝑀 be the number of males in the company, then 𝑀 is 40 percent of 𝑁. That is 40 out of 100 multiplied by 𝑁, which is 0.4𝑁.
Now, to find the total number of employees in the company, that’s uppercase 𝑁, we’re going to use the capture–recapture formula. This tells us that the population size uppercase 𝑁 is equal to uppercase 𝑃 multiplied by lowercase 𝑛 divided by lowercase 𝑝. This is where uppercase 𝑃 is the number initially captured, tagged, and released. So that’s the first sample. Lowercase 𝑛 is the number subsequently captured. So that’s the second sample. And lowercase 𝑝 is the number of the second sample. That’s the number out of the 𝑛 in the second sample, which are found to be tagged.
So how does this apply to our scenario of employees? Remember, we’re trying to find the total number of employees, that’s uppercase 𝑁. In our case, we’re going to identify all male employees, that’s uppercase 𝑀, which is 0.4 times 𝑁, as those captured, tagged, and released. That is uppercase 𝑃 in our capture–recapture formula. And so uppercase 𝑃 is equal to 0.4 times uppercase 𝑁.
Now, from the question, we know that our sample size, that is, the number subsequently captured, which is lowercase 𝑛 in our capture–recapture method, is 10. That’s 10 employees were selected. We’re told also that the males in this sample represent five percent of the males in the company. Now, recalling that we’re equating being male with being tagged, then lowercase 𝑝 — that’s the number in the second sample found to be tagged, that’s found to be male — is five percent of the total number of males in the company. And that’s 0.05 multiplied by 0.4 multiplied by 𝑁. Hence, lowercase 𝑝 is 0.02𝑁.
So we’re trying to find uppercase 𝑁. That’s the total number of employees in the company. We found that uppercase 𝑃 is 0.4𝑁. That’s uppercase 𝑁. Lowercase 𝑛 is 10. That’s the sample size. And lowercase 𝑝 is 0.02 times uppercase 𝑁.
So now, making some space for our calculation, we have uppercase 𝑁, that’s the total number of employees in the company, is equal to 0.4 multiplied by uppercase 𝑁 multiplied by 10 all divided by 0.2 again multiplied by uppercase 𝑁. Dividing both numerator and denominator on our right-hand side by uppercase 𝑁, we have our population size uppercase 𝑁 is equal to 0.4 multiplied by 10 all divided by 0.02. And this evaluates to 200. And so using the capture–recapture method, we find that the total number of employees in that company is 200.