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?

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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)