**(Short Version for those who want to provide a quick answer)**- I have 40 people I want to follow and record time spent performing tasks so I can say how long it takes to do each task. How many people do I follow and for how many days? I know that it all depends on what confidence level you want and what you define as statistically significant. Assume all of those variables are defined (if they are required in your explanation please just give example values). What are the formulas I can use to prove that I've met my criteria for confidence, significance, or any other factors that need to be addressed when analyzing a statistic?

**(Long Version for those who like a little background)**- I've been assigned to lead a time study at my work on our maintenance employees. The goal is to increase efficiency from a "lean" or "continuous improvement" perspective. This will include observing our technicians and documenting the time it takes for them to perform tasks. The end result would be to identify opportunities for improvement and implement solutions. For example, we may find out that it takes an average of 2 hours to change out a motor and 1 hour of that time is typically spent gathering parts and tools to perform the job. As a result we would look at ways to streamline gathering parts/tools so the job could be done quicker.

I'm having a difficult time buying off on the idea, however, because the logic for sample size selection seems fuzzy to me. We have 3 main departments, each of which have 12-15 technicians. I've been told to have a team do observations on 9 technicians from each department for 2 days each. This will mean a total of 54 shift observations (9 technicians/department x 3 departments x 2 days). I asked where these numbers for sample selection came from and I was told that it was just arbitrarily chosen by management because they felt "it would be enough".

I'm worried that the time and effort that my team and I will be putting into this study will be pointless because I have no confidence that those 54 shifts observed will be an accurate representation of my population. If we have a total of 40 technicians, and each one works 5 shifts a week, then over the course of a year there are 10,400 shifts worked. This comes out to around a 0.5% sample size if my population was for an entire year. So my question is this: How do I go about selecting my sample size for a study such as this? I know that it all depends on what confidence level you want and what you define as statistically significant. Assume all of those variables are defined (if they are required in your explanation please just give example values). What are the formulas I can use to prove that I've met my criteria for confidence, significance, or any other factors that need to be addressed when analyzing a statistic?

I appreciate any help that you can provide. Also, if you have any links to sites for better reference, please let me know. I don't mind reading, doing research, or digging for the information. I've just hit a wall when trying to find an answer to this particular issue and wouldn't waste everyone's time if I hadn't tried finding the answer already. Thanks!