Please help me what option and how?

Question

A carnival game is designed so that approximately 10% of players will win a large prize. If there is evidence that the percentage differs significantly from this target, then adjustments will be made to the game. To investigate, a random sample of 100 players is selected from the large population of all players. Of these players, 19 win a large prize. The question of interest is whether the data provide convincing evidence that the true proportion of players who win this game differs from 0.10. 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 1

The conditions for inference in a z-test for one proportion are:

1. Random Sample: The sample should be randomly selected from the population of interest.
2. Sample Size: Both np and n(1-p) should be greater than or equal to 10, where n is the sample size and p is the hypothesized proportion.
3. Independence: The sample observations should be independent of each other.

In this scenario, we are given that a random sample of 100 players is selected, so the first condition is met.

The expected count of players who win a large prize is np = 100 x 0.10 = 10 and the expected count of players who do not win a large prize is n(1-p) = 100 x 0.90 = 90. Both of these expected counts are greater than or equal to 10, so the second condition is also met.

Assuming that each player's outcome is independent of each other, the independence condition is also met.

Therefore, all the conditions for conducting a z-test for one proportion are met, and the correct answer is A: Yes, the random, 10%, and large counts conditions are all met.