A recent study suggests that 74% of high school students buy lunch. Olivia is a teacher at a high school who would like to know whether the percentage is less for high school students at her school. She randomly selects 50 students at her school and finds that 36 of the students buy lunch. Are all of the conditions for this hypothesis test met, and if so, what are the null and alternative hypotheses for this hypothesis test?
Select the correct answer below:


{H0:p≤0.74Ha:p>0.74

{H0:p≠0.74Ha:p=0.74

{H0:p≥0.74Ha:p<0.74

{H0:p=0.74Ha:p≠0.74


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.

A recent study suggests that 74% of high school students buy lunch. Olivia is a teacher at a high school who would like to know whether the percentage is less for high school students at her school. She randomly selects 50 students at her school and finds that 36 of the students buy lunch. Are all of the conditions for this hypothesis test met, and if so, what are the null and alternative hypotheses for this hypothesis test?

Explore a hypothesis test where a high school teacher wants to determine if fewer than 74% of students buy lunch. This page covers hypothesis testing for proportions, null and alternative hypotheses, and conditions for the test, including random sampling, independence, and normality conditions. Suitable for students in Grades 11-12.

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