Choose 1 item in a pair-- for 51 pairs. How do I get a certain number of "selected" responses for each of 102 items? I'm new here, so please let me know if I'm not following proper etiquette.
I’m doing a project that requires gathering judgements for 100 items. These hundred items will be randomly paired without repeats, for a total of 50 randomly chosen pairs (order chosen doesn't matter). Each participant in the project will see these 50 pairs, and will select one item in each pair as their response. Each item in the pairs will have roughly equal probability of being chosen.
My question is about how many participants I would need in order to get 40 "selection" judgements per item. In other words, I want 40 "chosen" responses for each of my 100 items. How many participants does that require?
Thank you in advance!
 A: It depends on how sure you want to be that an image is picked $40$ times.  If you imagine that each participant throws a fair coin to choose the image, for $n$ participants you will have each image selected on average $\frac n2$ times.  The standard deviation on this is $\frac 12\sqrt n$.  If you want a $95\%$ chance that a given image has been selected $40$ times, you need $40$ to be the mean minus $1.96$ standard deviations or $40=\frac n2-1.96\cdot \frac 12\sqrt n$.  Alpha tells us you need $100$ people under this model.  You will still probably have a few images selected less than $40$ times, so you might want more people.
A: Unless I am misunderstanding you, it sounds like each participant chooses $51$ images. Also, if you want, on average, each of the $102$ images to be chosen $40$ times, then that means you are looking for $102 \cdot 40 = 4080$ choices. So, you need $\frac{4080}{51}=80$ participants.
Put differently: with each image being chosen with roughly the same likelihood, you need two participant choices to get each image selected once. So, for an image to be selected $40$ times, you need $40 \cdot 2 = 80$ participants.
