Statistics is my nightmare

Ihatestatistics

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Dec 13, 2019
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Ok, so I consider myself a pretty decent at math, but statistics literally makes me want to throw my computer across the room. 8-)

That being said, I own a water sports company, and we have to administer random drug tests at a number equal to 50% of the total pool of employees at a minimum each year. We do this through a consortium. The total pool of people is 520. So each year 260 tests have to be administered. So each week 5 names must be pulled to get to the 260 total tests a year, and you have the same chance each week regardless of how many times you are picked. One of my employees has complained that he keeps getting called (not because he does drugs, hes a really good kid) but because its a hassle to drive there after works and deal with the whole process. So it got my brain going, and I quickly remembered why i hate statistics so much.

So if anyone could help me out with some formulas that i could use to figure some things out, I would appreciate it.

Being that 5 names are pulled each week for 52 weeks, with a total pool of 520...
Whats the probability of being picked one time a year, 2 times....ect
Whats the probability of being picked 2 times a year for 2 years, 3.....ect

Any help would be awesome, thanks.
 
I'll give you one for free. I'm bored, this morning.

1) Keep in mind that Statistics is not a branch of mathematics. It just requires mathematics to deal with it.
2) Not all statistics is the same. Sometimes, you're just counting things.
3) You can probably employ a Poisson Distribution to solve your problem to your satisfaction. It is nicely scalable. It may not be perfectly accurate or appropriate, but it may be useful if we just ignore the tail.
4) Have you considered compensating the employees for their time and effort required for the testing? Maybe nothing for a first draw, but popping up some benefit for a second draw, and scaling up from there for any subsequent draw in the same year.

p(being chosen in one week) = 5/520 = 1/104 = 0.009615

One Week
0 - 99.04% or 515 people not selected.
1 - 0.95% or 5 people selected.
2 - Theoretically, there us a 0.05% chance of being selected twice. This can't happen, but we'll need to get used to it to benefit from the Poisson simplification.

One Month (A little more than 4 weeks)
0 - 96.07% or 500 people not selected.
1 - 3.85% or 20 people selected once.
2 - 0.08% or not very likely to be selected twice.
Negligible probabilities after that. Theoretically, someone could be selected four times. This probability is very low. Remember that these are EXPECTATIONS, not mandates.

One Year (52 weeks)
0 - 60.65% or 315 people not selected.
1 - 30.33% or 158 people selected once.
2 - 7.58% or 39 people selected twice.
3 - 1.26% or 7 people selected thrice
4 - 0.16% or 1 poor person selected four times.
Negligible probabilities after that.
This may indicate that you are testing only around 205 different people per year. Is your procedure adequate?

Two Years (104 weeks)
0 - 36.79% or 191 people not selected.
1 - 36.79% or 191 people selected once.
2 - 18.39% or 96 people selected twice.
3 - 6.13% or 32 people selected thrice
4 - 1.53% or 8 people selected four times
5 - 0.31% or 2 people abused by 5 selections
Negligible probabilities after that.

Three Years (156 weeks)
0 - 22.31% or 116 people not selected.
1 - 33.47% or 174 people selected once.
2 - 25.10% or 131 people selected twice.
3 - 12.55% or 65 people selected thrice
4 - 4.71% or 24 people selected four times
5 - 1.41% or 7 people selected five times
6 - 0.35% or 2 people who likely will not come back to work after being targeted 6 times.
Negligible probabilities after that.
This one says that 486 people are selected an average of once per year or less often. That's not bad.
It also says that 34 people are selected an average of more than once per year. You'll have to decide what to do about this irritation.

Not all these considerations are particularly rational. You'll have to decide on some things. In particular, what is the expected length of employment in your business? If it's only two years or so, why worry about the 3-year effect? Many other factors could influence your process; it's not all about numbers.

Alway remember, you get what you pay for! :)
 
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Thank you for your time. As for the pay, my employees make 50k a year without the need for any degree or special skill, so they will be alright 8-)

The issue I was trying to resolve was that the testing must be done by a “scientifically proven method” for randomness. So I wanted to see how unlikely it truly is to be picked so many times, so when I can the consortium and tell them that something isn’t right, I atleast have some days to back that up.
 
Fair enough. I think we got that far.

How do you randomize your selection?
 
I am required by USCG law to implement a DOT approved drug testing program. Being that I also am a Captain, I can not run the program as the owner. Say I was the owner with 10 employees, and I didn’t actually work on the boats. Then I could implement my own drug testing program, and pick pick 5 random employees a year, to satisfy the requirements. Being that I am also a active captain at the business, I have to go through a drug testing consortium, and my employees are added to my account at the same consortium. So they don’t pay for anything as long as the work for 6 months without quitting of being fired for cause, we pay for it all. So the consortium chooses who takes the random drug tests, by DOT standards, and US law they must choose the randoms with a “scientifically proven method”. It’s seemed to me the same person being chosen 2 time a year for 3 years l, when on average it should be closer to .59 times a year was a statistically very unlikely. Then I reviewed all the tests for my company over the last 3 years and it seems we are being tested at a number that far exceeds the statistical norm. I just am really bad at statistics for some reason, my brain doesn’t like the formulas or wording. So I figured somebody here could help me out, so when I go talk to the consortium I can show them the numbers and ask how exactly they choose the numbers, and why we are being tested at such a high rate.
 
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