Question

Give me some questions with rejecting the null, accepting the null, and failure to reject null with H_0 and H_a (show the answers at the bottom) and incorporate >,<, more than equal, less than equal, equal or not, and =

Answer 1

Here are some hypothesis testing questions that involve rejecting the null hypothesis \(H_0\) or failing to reject the null hypothesis, along with their respective alternatives \(H_a\). The answers are provided at the bottom:

### Question 1
**Question:** A researcher wants to test if a new drug is more effective than the existing medication. The null hypothesis is that the new drug is not more effective, while the alternative hypothesis suggests it is more effective.

- \(H_0\): The new drug is ≤ as effective as the existing drug.
- \(H_a\): The new drug is > as effective than the existing drug.

**Conclusion:** If the p-value is 0.03 and the significance level (α) is 0.05, what is the decision regarding the null hypothesis?

### Question 2
**Question:** A quality control engineer inspects a batch of light bulbs to determine if the average lifespan of bulbs is equal to 1000 hours.

- \(H_0\): The average lifespan of bulbs = 1000 hours.
- \(H_a\): The average lifespan of bulbs ≠ 1000 hours.

**Conclusion:** If the test statistic falls within the critical values at α = 0.01, what is the decision regarding the null hypothesis?

### Question 3
**Question:** A teacher wants to investigate if the students taught with an interactive method have scores that are less than those taught by traditional methods.

- \(H_0\): The average score of students with the interactive method ≥ the average score of students with the traditional method.
- \(H_a\): The average score of students with the interactive method < the average score of students with the traditional method.

**Conclusion:** If the p-value is 0.07 and α = 0.05, what is the decision regarding the null hypothesis?

### Question 4
**Question:** A manufacturer claims that their batteries last at least 500 hours. A consumer group tests the batteries to see if they live up to this claim.

- \(H_0\): The average lifespan of the batteries < 500 hours.
- \(H_a\): The average lifespan of the batteries ≥ 500 hours.

**Conclusion:** If the p-value is 0.01 and α = 0.05, what is the decision regarding the null hypothesis?

### Question 5
**Question:** A company claims that the average time it takes to process an order is equal to 2 hours. A competitor believes the average time is less than that.

- \(H_0\): The average processing time = 2 hours.
- \(H_a\): The average processing time < 2 hours.

**Conclusion:** If the critical value at α = 0.05 is not exceeded, what is the decision regarding the null hypothesis?

---

### Answers

1. **Conclusion:** Reject \(H_0\) (Because p-value < α)

2. **Conclusion:** Fail to reject \(H_0\) (Because the test statistic falls within the critical values)

3. **Conclusion:** Fail to reject \(H_0\) (Because p-value > α)

4. **Conclusion:** Reject \(H_0\) (Because p-value < α)

5. **Conclusion:** Fail to reject \(H_0\) (Because the critical value is not exceeded)