One condition for performing a hypothesis test is that the observations are independent.  Mark is going to take a sample from a population of 400 students.  

**How many students will Mark have to sample without replacement to treat the observations as independent?**

When sampling without replacement, the rule of thumb for treating observations as independent is that the sample size should not exceed 10% of the total population. This is known as the "10% condition."

Given:
- Population size: \( N = 400 \)

To maintain the independence assumption, the sample size \( n \) must satisfy:

\[
n \leq 0.10 \times N
\]

Substituting the given population size:

\[
n \leq 0.10 \times 400 = 40
\]

Thus, Mark can sample up to **40 students** without replacement and still reasonably treat the observations as independent.

Would you like further details or explanations on this? 

Here are some related questions for further exploration:
1. Why is the 10% condition important in hypothesis testing?
2. How does sampling without replacement affect hypothesis testing assumptions?
3. What other conditions are necessary for a hypothesis test besides independence?
4. What are the differences between sampling with and without replacement?
5. How would increasing the population size affect the sample size for independence?

**Tip:** Always check whether other assumptions (e.g., normality, randomness) are met in addition to the 10% condition when performing a hypothesis test.

One condition for performing a hypothesis test is that the observations are independent. Mark is going to take a sample from a population of 400 students. How many students will Mark have to sample without replacement to treat the observations as independent?

Find out how many students Mark can sample from a population of 400 to maintain independence in hypothesis testing. This problem explains the 10% condition for sampling without replacement, providing clear guidance on how to maintain statistical assumptions.

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