In this scenario, Olivia wants to determine if the percentage of students at her school who buy lunch is less than 74%. The correct hypothesis test would aim to assess whether the proportion at her school is lower than the 74% stated in the general study.

Let's break it down:

• The null hypothesis $H_0$ represents the assumption that the proportion of students at her school who buy lunch is at least 74%. This is based on the general assumption provided by the study.
• The alternative hypothesis $H_a$ represents the suspicion or claim Olivia wants to test: that the proportion of students at her school who buy lunch is less than 74%.

Thus, the correct hypotheses are:

• $H_0$: $p \geq 0.74$ (the proportion of students who buy lunch at her school is at least 74%)
• $H_a$: $p < 0.74$ (the proportion of students who buy lunch at her school is less than 74%)

This matches the third option:

$H_0: p \geq 0.74$ and $H_a: p < 0.74$

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Conditions for the hypothesis test:

There are three primary conditions for a proportion hypothesis test:

1. Random sample: The sample must be randomly selected. Since Olivia randomly selected the students, this condition is met.
2. Independence: Each observation should be independent. This is typically satisfied if the sample size is small compared to the population (less than 10% of the population), or there is no direct relationship between the students. Assuming Olivia's school has at least 500 students (so 50 students is less than 10%), this condition is likely met.
3. Normality: The sample size must be large enough such that $np$ and $n(1 - p)$ are both greater than or equal to 10. Let's check:
   • $np = 50 \times 0.74 = 37 \geq 10$
   • $n(1 - p) = 50 \times (1 - 0.74) = 50 \times 0.26 = 13 \geq 10$

Since both conditions are met, the normality condition is also satisfied.

Therefore, all conditions for the hypothesis test are met.

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Would you like details on how the hypothesis test can be conducted?

Related questions:

1. How do you calculate the test statistic for a proportion hypothesis test?
2. What is a p-value, and how is it interpreted in a hypothesis test?
3. How can we check the normality condition for a smaller sample size?
4. What are Type I and Type II errors in hypothesis testing?
5. How does changing the sample size affect the results of a hypothesis test?

Tip: In hypothesis testing, always carefully define the null and alternative hypotheses based on the specific claim being investigated.