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. the computer output gives the results of a z - test for one proportion. what conclusion should be made at the a = 0.05 level? because the p - value < a = 0.05, there is convincing evidence that the true proportion of players who win this game differs from 0.10. because the p - value < a = 0.05, there is convincing evidence that the true proportion of players who win this game differs from 0.19. because the p - value < a = 0.05, there is not convincing evidence that the true proportion of players who win this game differs from 0.10. because the p - value < a = 0.05, there is not convincing evidence that the true proportion of players who win this game differs from 0.19.

Explanation:

In hypothesis testing, if the P - value is less than the significance level ($\alpha$), we reject the null hypothesis. Here, the null hypothesis is $H_0:p = 0.1$ (where $p$ is the proportion of players who win the game) and the alternative hypothesis is $H_a:p
eq0.1$. The significance level $\alpha = 0.05$ and the P - value is $0.0027$. Since $0.0027<0.05$, we reject the null hypothesis. The claim in the null hypothesis is about the proportion being $0.1$, not $0.19$.

Answer:

Because the $P$-value $<\alpha = 0.05$, there is convincing evidence that the true proportion of players who win this game differs from $0.10$.