## Testing a learning curve

Hi all,

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%