A teacher has a large container of blue, red, and green beads. She reports to the students that the proportion of blue beads in the container is 0.30. The students feel the proportion of blue beads is lower than 0.30. A student randomly selects 50 beads and finds that 12 of the beads are blue. Which hypotheses would test the students’ claim?

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.

Explore how to conduct a hypothesis test for the proportion of blue beads in a container. Given a student's claim that the proportion is lower than 0.30, this problem covers key concepts in hypothesis testing, including formulating null and alternative hypotheses, calculating the test statistic, and interpreting the results. Suitable for Grades 11-12 or college-level statistics students.

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