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?

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

To determine whether the conditions for inference are met for conducting a z-test for one proportion, we need to check the following conditions:

1. **Random condition**: The sample must be randomly selected. In this case, a random sample of 100 players is selected, so this condition is met.

2. **10% condition**: The sample size should be no more than 10% of the population. Assuming that the population of all carnival game players is much larger than 100, this condition is met.

3. **Large counts condition**: Both \( np \) and \( n(1 - p) \) must be at least 10, where \( p \) is the hypothesized population proportion (in this case, 0.10), and \( n \) is the sample size (100). 

    - \( np = 100 \times 0.10 = 10 \)
    - \( n(1 - p) = 100 \times (1 - 0.10) = 100 \times 0.90 = 90 \)

   Since both 10 and 90 are greater than or equal to 10, the large counts condition is also met.

Thus, **all three conditions (random, 10%, and large counts) are met**, so the correct answer is:

**Yes, the random, 10%, and large counts conditions are all met.**

Would you like further details on conducting the z-test?

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Here are 5 related questions:
1. How do you calculate the z-statistic for a one-proportion z-test?
2. What is the null hypothesis for this one-proportion z-test?
3. How do you interpret the p-value in this context?
4. What is the conclusion if the p-value is less than the significance level?
5. How would increasing the sample size affect the conditions for the test?

**Tip:** Always ensure the population is large enough when using the 10% condition to avoid bias in sampling!

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?

Determine if the conditions for conducting a one-proportion z-test are met using a sample of 100 players from a carnival game. This problem reviews the random, 10%, and large counts conditions, using the example of 19 winners from 100 players with a 10% hypothesized proportion. Suitable for college-level statistics students.

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