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According to a soccer coach, 75% of soccer players have had at least one sprained ankle. An athletic trainer would like to investigate this claim. To do so, the trainer selects a random sample of 125 college soccer players from across the country and finds that 99 of them have had at least one sprained ankle.

Are the conditions for inference met for conducting a z-test for one proportion?

A. Yes, the random, 10%, and large counts conditions are all met.
B. No, the random condition is not met.
C. No, the 10% condition is not met.
D. No, the large counts condition is not met.

Answer :

After evaluating the random, 10%, and large counts conditions for conducting a z-test for one proportion, all conditions are satisfied. Thus, the athletic trainer can proceed with the z-test.

The student's question is whether the conditions for inference are met for conducting a z-test for one proportion based on the information that a soccer coach claims 75% of players have sprained an ankle at least once, and an athletic trainer finds that 99 out of 125 college soccer players have had at least one sprained ankle. To answer this question, we check the conditions as follows:

  • Random condition: The sample must be random. Assuming the trainer selected the players at random, this condition is met.
  • 10% condition: The sample size must not exceed 10% of the population. Since there is likely more than 1250 college soccer players in the country, this condition is satisfied.
  • Large counts condition: We need to check if np and n(1-p) are both ≥ 10. For the claim that 75% of players have had a sprained ankle, np = 125(0.75) = 93.75 and n(1-p) = 125(0.25) = 31.25. Both are greater than 10, so this condition is also met.

Since all conditions are met, the correct answer is a. Yes, the random, 10%, and large counts conditions are all met.

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