I recently read this passage from a website and I just can't work out the math.
Overall, it says you can be 93.75% confident of having the true median parameter within an interval, obtained from a random sample of 5 out of a 10 000 population.
Could someone guide me to obtain this value? Here's the original passage:
Pretend for a moment that you’re a decision-maker for a large corporation with 10,000 employees. You’re considering automating part of some routine activity, like scheduling meetings or preparing status reports. But you are facing a lot of uncertainty and you believe you need to gather more data. Specifically, one thing you’re looking for is how much time the typical employee spends each day commuting.
How would you gather this data?
You could create what essentially would be a census where you survey each of the 10,000 employees. But that would be very labor-intensive and costly. You probably wouldn’t want to go through that kind of trouble. Another option is to get a sample, but you are unsure what the sample size should be to be useful.
What if you were told that you might get enough information to make a decision by sampling just five people?
Let’s say that you randomly pick five people from your company. Of course, it’s hard for humans to be completely random, but let’s assume the picking process was about as random as you can get.
Then, let’s say you ask these five people to give you the total time, in minutes, that they spend each day in this activity. The results come in: 30, 60, 45, 80, and 60 minutes. From this, we can calculate the median of the sample results, or the point at which exactly half of the total population (10,000 employees) is above the median and half is below the median.
Is that enough information?
Many people, when faced with this scenario, would say the sample is too small – that it’s not “statistically significant.” But a lot of people don’t know what statistically significant actually means.
Let’s go back to the scenario. What are the chances that the median time spent in this activity for 10,000 employees, is between 30 minutes and 80 minutes, the low and high ends, respectively, of the five-employee survey?
When asked, people often say somewhere around 50%. Some people even go as low as 10%. It makes sense, after all; there are 10,000 employees and countless individual commute times in a single year. How can a sample that is viewed as not being statistically significant possibly get close?
Well, here’s the answer: the chances that the median time spent of the population of 10,000 employees is between 30 minutes and 80 minutes is a staggering 93.75%.
In other words, you can be very confident that the median time spent is between 30 minutes and 80 minutes, just by asking five people out of 10,000 (or 100,000, or 1,000,000 – it’s all the same math).