The problem presents a situation where the students want to test whether the proportion of blue beads is lower than what the teacher claims, which is 0.30 (or 30%).

To set up the hypotheses for this test, we use the following standard form for hypothesis testing:

• Null hypothesis (H₀): The proportion of blue beads in the container is equal to the teacher's claimed proportion (0.30).

• Alternative hypothesis (H₁): The proportion of blue beads in the container is less than the teacher's claimed proportion (0.30), since the students believe the proportion is lower.

Thus, the hypotheses would be:

• Null hypothesis (H₀): $p = 0.30$
• Alternative hypothesis (H₁): $p < 0.30$

Where $p$ represents the true proportion of blue beads in the population.

This is a one-tailed test because the alternative hypothesis suggests that the proportion is specifically less than 0.30.

Would you like details on how to proceed with this hypothesis test, such as calculating a test statistic or finding a p-value?

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Here are 5 related questions that expand on this problem:

1. How do you calculate the test statistic for a one-sample proportion test?
2. What is the significance level, and how does it affect the hypothesis test result?
3. How can we interpret a p-value in the context of this hypothesis test?
4. How do you construct a confidence interval for the proportion of blue beads?
5. What assumptions need to be met for the results of a proportion hypothesis test to be valid?

Tip: In hypothesis testing, always carefully determine whether a test is one-tailed or two-tailed based on the wording of the research question or claim.