Suppose someone kicks a football at a goal. However, I want to test to see whether or not practice improves technique - whether there is a "learning curve". I get 100 people to kick footballs at a goal 50 times each.

I want to test whether there really is a learning curve, or whether there is some statistical effect - someone has suggested that results should follow the binomial (or perhaps another?) distribution and this has a built-in learning curve.

First question: the binomial distribution can only be used when trials are independent and the probability of success is constant throughout the experiment, therefore surely the binomial distribution cannot be used if a learning curve is suspected?

Second question: Could there be some other statistical effect going on i.e. a learning curve inherent in statistics?

Third question: how do I derive a learning curve equation and test whether that learning curve is statistically significant?

Here's the data:

Trial Success rate

1 43%

2 50%

3 62%

4 52%

5 61%

6 61%

7 63%

8 59%

9 60%

10 50%

11 60%

12 62%

13 63%

14 70%

15 61%

16 71%

17 75%

18 80%

19 77%

20 70%

21 70%

22 74%

23 79%

24 71%

25 67%

26 74%

27 70%

28 70%

29 70%

30 65%

31 81%

32 79%

33 81%

34 94%

35 76%

36 82%

37 88%

38 87%

39 67%

40 71%

41 82%

42 88%

43 82%

44 57%

45 82%

46 92%

47 93%

48 77%

49 77%

50 85%