Answer :
When addressing a hypothesis testing question like this one, it's important to consider the context and what you are trying to prove or disprove. Here is how to approach the question step by step:
1. Understand the Context:
- You have a specific breed of sheep dog with an average resting heart rate of 115 beats per minute (bpm) according to a manual.
- A farmer believes that his dog's heart rate is slower because the dog is acting lethargic.
2. Identify the Problem:
- You want to test if the dog's average heart rate is indeed slower than the normal rate for its breed (115 bpm).
3. Set Up Hypotheses:
- The null hypothesis ([tex]\(H_0\)[/tex]) is usually a statement of no effect or no difference. In this case, it would be that the dog’s heart rate is 115 bpm, which represents the standard or expected rate.
- The alternative hypothesis ([tex]\(H_A\)[/tex]) should capture the suspicion or the claim to be tested. Here, the farmer suspects the heart rate is lower than normal, so you would state that the dog's heart rate is less than 115 bpm.
4. Choose the Correct Option:
- From the description of hypotheses:
- [tex]\(H_0: \mu = 115\)[/tex]: The notation [tex]\(\mu\)[/tex] represents the true mean heart rate of the farmer's dog over all occasions, assumed to be 115 bpm under the null hypothesis.
- [tex]\(H_A: \mu < 115\)[/tex]: The alternative hypothesis that the mean heart rate is less than 115 bpm, reflecting the farmer's suspicion.
5. Conclusion:
- The appropriate hypothesis test reflecting the farmer's belief (that the dog’s heart rate is slower) would be to use:
- Option C: [tex]\(H_0: \mu = 115, \quad H_A: \mu < 115\)[/tex]
This methodical breakdown shows how you determine that the correct set of hypotheses to test the belief that the dog's heart rate is slower than the average is indeed option C.
1. Understand the Context:
- You have a specific breed of sheep dog with an average resting heart rate of 115 beats per minute (bpm) according to a manual.
- A farmer believes that his dog's heart rate is slower because the dog is acting lethargic.
2. Identify the Problem:
- You want to test if the dog's average heart rate is indeed slower than the normal rate for its breed (115 bpm).
3. Set Up Hypotheses:
- The null hypothesis ([tex]\(H_0\)[/tex]) is usually a statement of no effect or no difference. In this case, it would be that the dog’s heart rate is 115 bpm, which represents the standard or expected rate.
- The alternative hypothesis ([tex]\(H_A\)[/tex]) should capture the suspicion or the claim to be tested. Here, the farmer suspects the heart rate is lower than normal, so you would state that the dog's heart rate is less than 115 bpm.
4. Choose the Correct Option:
- From the description of hypotheses:
- [tex]\(H_0: \mu = 115\)[/tex]: The notation [tex]\(\mu\)[/tex] represents the true mean heart rate of the farmer's dog over all occasions, assumed to be 115 bpm under the null hypothesis.
- [tex]\(H_A: \mu < 115\)[/tex]: The alternative hypothesis that the mean heart rate is less than 115 bpm, reflecting the farmer's suspicion.
5. Conclusion:
- The appropriate hypothesis test reflecting the farmer's belief (that the dog’s heart rate is slower) would be to use:
- Option C: [tex]\(H_0: \mu = 115, \quad H_A: \mu < 115\)[/tex]
This methodical breakdown shows how you determine that the correct set of hypotheses to test the belief that the dog's heart rate is slower than the average is indeed option C.
