Answer :
To solve this problem, we'll need to set up a hypothesis test to determine if the farmer's observation that his sheep dog's heart rate is slower than the breed's average is statistically valid.
1. Understand the Problem:
- The average resting heart rate for this type of sheep dog is 115 beats per minute (bpm), according to the Merck Veterinary Manual.
- The farmer measured his dog's heart rate 15 times, finding an average rate of 110.2 bpm.
- We want to test if the heart rate is significantly less than 115 bpm, which suggests a one-tailed test.
2. Define the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): This is the statement that there is no effect or no difference. In this context, it means the dog's average heart rate is equal to the standard breed average.
[tex]\[
H_0: \mu = 115
\][/tex]
- Alternative Hypothesis ([tex]\(H_A\)[/tex]): This statement reflects the farmer's belief. He thinks the dog's heart rate is slower, so the alternative hypothesis should suggest the mean heart rate is less than 115 bpm.
[tex]\[
H_A: \mu < 115
\][/tex]
3. Choose the Correct Answer Based on the Hypotheses:
From the options provided:
- Option A: [tex]\(H_0: \mu = 115, \quad H_A: \mu < 115\)[/tex]
This directly matches our hypotheses setup, indicating we're conducting a one-tailed test where the alternative hypothesis is that the mean heart rate of the dog is lower than 115 bpm.
Hence, the correct choice is:
A. [tex]\(H_0: \mu = 115, \quad H_A: \mu < 115\)[/tex]
This setup suits the farmer's suspicion that his dog's heart rate might be slower than the breed average.
1. Understand the Problem:
- The average resting heart rate for this type of sheep dog is 115 beats per minute (bpm), according to the Merck Veterinary Manual.
- The farmer measured his dog's heart rate 15 times, finding an average rate of 110.2 bpm.
- We want to test if the heart rate is significantly less than 115 bpm, which suggests a one-tailed test.
2. Define the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): This is the statement that there is no effect or no difference. In this context, it means the dog's average heart rate is equal to the standard breed average.
[tex]\[
H_0: \mu = 115
\][/tex]
- Alternative Hypothesis ([tex]\(H_A\)[/tex]): This statement reflects the farmer's belief. He thinks the dog's heart rate is slower, so the alternative hypothesis should suggest the mean heart rate is less than 115 bpm.
[tex]\[
H_A: \mu < 115
\][/tex]
3. Choose the Correct Answer Based on the Hypotheses:
From the options provided:
- Option A: [tex]\(H_0: \mu = 115, \quad H_A: \mu < 115\)[/tex]
This directly matches our hypotheses setup, indicating we're conducting a one-tailed test where the alternative hypothesis is that the mean heart rate of the dog is lower than 115 bpm.
Hence, the correct choice is:
A. [tex]\(H_0: \mu = 115, \quad H_A: \mu < 115\)[/tex]
This setup suits the farmer's suspicion that his dog's heart rate might be slower than the breed average.
