Question

In: Statistics and Probability

GOALKEEPER JUMP Left Center Right Kick to Left 52 5 45 Kick to Center 39 12...

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)

Solutions

Expert Solution

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.


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