In: Statistics and Probability
GOALKEEPER JUMP
Left | Center | Right | |
Kick to Left | 52 | 5 | 45 |
Kick to Center | 39 | 12 | 35 |
Kick to Right | 50 | 5 | 59 |
In soccer, serious fouls result in a penalty kick with one kicker and one defending goalkeeper. The accompanying table summarizes results from 302 kicks during games among top teams. In the table, jump direction indicates which way the goalkeeper jumped, where the kick direction is from the perspective of the goalkeeper. Use a 0.01 significance level to test the claim that the direction of the kick is independent of the direction of the goalkeeper jump. Do the results support the theory that because the kicks are so fast, goalkeepers have no time to react, so the directions of their jumps are independent of the directions of the kicks? (round to 3 decimals as needed)
Alpha, = 0.01
Null hypothesis
H0: direction of the kick is independent of the direction of the
goalkeeper jump
Alternative hypothesis
H1: direction of the kick is not independent of the direction of
the goalkeeper jump
observed frequencies,f0:
Left | Center | Right | Total | |
Kick to Left | 52 | 5 | 45 | 102 |
Kick to Center | 39 | 12 | 35 | 86 |
Kick to Right | 50 | 5 | 59 | 114 |
Total | 141 | 22 | 139 | 302 |
expected frequencies,fe:
fe = (row total * column total) / total
for example:
fe1 = 102*141/302 = 47.6225
fe5 = 86*22/302 = 6.265
Right | Center | Left | |
Kick to Right | 47.6225 | 7.4305 | 46.947 |
Kick to Center | 40.1523 | 6.265 | 39.5828 |
Kick to Left | 53.225 | 8.305 | 52.4702 |
f_0 | f_e | (f_0-f_e)^2 | (f0-fe)^2 / fe |
52 | 47.6225 | 19.1625 | 0.4024 |
5 | 7.4305 | 5.9073 | 0.7950 |
45 | 46.947 | 3.7908 | 0.0807 |
39 | 40.1523 | 1.3278 | 0.0331 |
12 | 6.265 | 32.8902 | 5.2498 |
35 | 39.5828 | 21.0020 | 0.5306 |
50 | 53.225 | 10.4006 | 0.1954 |
5 | 8.305 | 10.9230 | 1.3152 |
59 | 52.4702 | 42.6383 | 0.8126 |
Sum | 9.4148 |
test statistic,X^2 = 9.4148
degree of freedom = (number of rows - 1) * (number of columns -
1) [not including total]
df = (3-1)*(3-1) = 4
P_value = 0.0007
Since P_value < 0.05, hence reject H0.
We can conclude that the direction of the kick is not independent of the direction of the goalkeeper jump.