To assess which condition for inference has not been met, let's break down each statement:

Condition I: The data are collected using a random sampling method.

• The professor surveys the first 25 students who show up for the exam. This is not a random sampling method because the students who arrive first are a specific subset of the population, which could be biased. The students who show up early might not be representative of the entire class (e.g., they might have studied more or less than the rest of the students).

Condition II: The sample size is large enough to assume the normality of the distribution of sample means.

• A sample size of 25 students can be considered sufficiently large for the Central Limit Theorem (CLT) to apply, assuming the underlying distribution of study hours is not extremely skewed. According to the CLT, sample means tend to follow a normal distribution when the sample size is large enough (usually n ≥ 30 is a common rule of thumb, but 25 is often considered adequate depending on the population's distribution).

Condition III: The sample size is less than 10% of the population size.

• The sample size (25) is much less than 10% of the population size (525). $\frac{25}{525} \approx 4.76\%$, which is clearly less than 10%, so this condition is met.

Conclusion:

Condition I has not been met because the sampling method is not random.

Answer: I only

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Would you like more details on this topic? Or perhaps further clarification?

Related questions:

1. Why is random sampling important in statistical inference?
2. How does the Central Limit Theorem help in estimating population parameters?
3. What other methods could the professor use to select a random sample from the students?
4. What happens if the sample size is too small for the Central Limit Theorem to apply?
5. How can you assess whether the population distribution is skewed?

Tip: When working with statistical inference, always ensure your sample is representative of the entire population to avoid bias in your estimates.