Statistical test for one sample proportions I am carrying out a study into the handedness of racehorses. There are clockwise and anti-clockwise racecourses. I want to test to see if 1) horses are more likely to win if they are left handed on anti-clockwise (left hand) courses than those who are right handed and 2) if they are more likely to win  if they are right handed on clockwise (right hand) courses than those who are left handed. Would the one sample z - test for proportions be appropriate. I have looked at 280 races so there are 280 winners in all. Thanks, I look forward to hearing your thoughts.
 A: As you are looking into the winners, the numbers lend themselves to be displayed in a 4 fields contingency table, which usually calls for a chi-square test for independence or a Fisher test.
However, you should consider, whether your data are i. i. d. enough. If there is one horse, that's left-handed and wins a lot of anti-clockWise courses than that's not the same, as if a lot of left-handed horses win a lot of anti-clockwise races.
edit: This may be your contingency table
            is left | is right
won clockw.    84       60
won antic.     71       65

A: As opposed to restricting to just one handedness of the course, another possibility is to measure wins and losses where group A is horses of same-handedness of course versus group B being horses with different-handedness of course. But you may want to use mantel-haenszel statistic where you stratify upon the handedness of the course.
I highly caution against making any causal claims though: Handedness of horse is likely inherited from parents, as is some degree of speed or racing expertise. On top of that, stud horses are presumably chosen because they have good racing qualities, perhaps handedness being one of them. So this isn't a randomized trial where handedness of horse is randomly assigned to each horse.
I'm not sure how it works, but it might be interesting to see the handedness of participating horses by handedness of race track - is there a sizeable difference in the number of lefthanded horses running on clockwise versus counterclockwise tracks? That might indicate whether there could be motivation to submit one's horse to a track with handedness that would give (perceived) advantage.
Also, +1 to @bernhard for pointing out that unless each horse appears only once in the dataset then there might be some issues with i.i.d. data
