Question

a penalty kick in soccer involves two players from different teams, the shooter and the goalie. during the penalty kick the shooter will try to score a goal by kicking a soccer ball to the left or right of the goal area. to prevent the shooter from scoring a goal, the goalie will move to the left or right of the goal area. the following table summarizes the directions taken by the shooter and the goalie for 372 penalty kicks.

	shooter
goalie move left	80	132	212
goalie move right	85	75	160
total	165	207	372

which of the following indicates an association between the shooters choice of direction and the goalies choice of direction?

a the marginal relative frequencies for the shooter and the goalie are equal.

b the marginal relative frequencies for the shooter and the goalie are not equal.

c the row totals are not equal.

d for the goalie, the relative frequency of a direction is equal to the relative frequency conditioned on the shooters direction.

e for the goalie, the relative frequency of a direction is not equal to the relative frequency conditioned on the shooters direction.

Explanation:

Step1: Recall the concept of association

In a two - way table, an association between two categorical variables (here, shooter's direction and goalie's direction) exists if the conditional relative frequencies are different from the marginal relative frequencies.

Step2: Analyze each option

• Option A: Marginal relative frequencies being equal does not imply association. It is about the overall distribution of each variable separately, not about their relationship.
• Option B: Marginal relative frequencies not being equal also does not directly imply association. It just shows different overall distributions of the two variables.
• Option C: Row totals not being equal is not a measure of association between the two variables.
• Option D: If the relative frequency of a goalie's direction is equal to the relative frequency conditioned on the shooter's direction, it means there is no association.
• Option E: If for the goalie, the relative frequency of a direction is not equal to the relative frequency conditioned on the shooter's direction, it indicates an association between the shooter's choice of direction and the goalie's choice of direction.

Answer:

E. For the goalie, the relative frequency of a direction is not equal to the relative frequency conditioned on the shooter's direction.