Answer :
To determine the correct null hypothesis ([tex]\(H_0\)[/tex]) and alternative hypothesis ([tex]\(H_A\)[/tex]) for the given scenario, let's break down the information:
1. Context and Average: The average resting heart rate for this breed of sheep dog is 115 beats per minute (bpm).
2. Observation and Belief: The farmer notices that his sheep dog seems more lethargic and suspects that her heart rate is slower than normal for her breed.
3. Sample Observation: The farmer took 15 measurements of the dog's heart rate and found the sample mean to be 110.2 bpm.
4. Hypothesis Testing Framework: The null hypothesis usually states that there is no change or no effect, which implies that the sheep dog's heart rate is the same as the average, 115 bpm. The alternative hypothesis reflects the farmer's belief that the heart rate is slower, meaning it is less than 115 bpm.
Based on this information, we can conclude that:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The mean heart rate of the farmer's sheep dog is equal to 115 bpm, or [tex]\(H_0: \mu = 115\)[/tex].
- Alternative Hypothesis ([tex]\(H_A\)[/tex]): The mean heart rate of the farmer's sheep dog is less than 115 bpm, or [tex]\(H_A: \mu < 115\)[/tex].
Therefore, the correct option that represents these hypotheses is:
C. [tex]\(H_0: \mu = 115, \quad H_A: \mu < 115\)[/tex]
1. Context and Average: The average resting heart rate for this breed of sheep dog is 115 beats per minute (bpm).
2. Observation and Belief: The farmer notices that his sheep dog seems more lethargic and suspects that her heart rate is slower than normal for her breed.
3. Sample Observation: The farmer took 15 measurements of the dog's heart rate and found the sample mean to be 110.2 bpm.
4. Hypothesis Testing Framework: The null hypothesis usually states that there is no change or no effect, which implies that the sheep dog's heart rate is the same as the average, 115 bpm. The alternative hypothesis reflects the farmer's belief that the heart rate is slower, meaning it is less than 115 bpm.
Based on this information, we can conclude that:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The mean heart rate of the farmer's sheep dog is equal to 115 bpm, or [tex]\(H_0: \mu = 115\)[/tex].
- Alternative Hypothesis ([tex]\(H_A\)[/tex]): The mean heart rate of the farmer's sheep dog is less than 115 bpm, or [tex]\(H_A: \mu < 115\)[/tex].
Therefore, the correct option that represents these hypotheses is:
C. [tex]\(H_0: \mu = 115, \quad H_A: \mu < 115\)[/tex]
