College

According to the Merck Veterinary Manual, the average resting heart rate for a certain type of sheepdog is 115 beats per minute (bpm). A Montana farmer notices his aging sheepdog has been acting more lethargic than usual. He believes that her heart rate is slower than normal for her breed. The farmer measured the dog's heart rate on 15 occasions and found a sample mean heart rate of 110.2 bpm. (Here, [tex]$\mu$[/tex] represents the mean heart rate of the farmer's sheepdog over all occasions.)

A. [tex]$H_0: \mu=115, \quad H_A: \mu\ \textless \ 115$[/tex]

B. [tex]$H_0: \mu=110.2, \quad H_A: \mu\ \textless \ 110.2$[/tex]

C. [tex]$H_0: \bar{x}=115, \quad H_A: \bar{x}\ \textgreater \ 115$[/tex]

D. [tex]$H_0: \mu=115, \quad H_A: \mu \neq 115$[/tex]

E. [tex]$H_0: \mu=115, \quad H_A: \mu\ \textgreater \ 115$[/tex]

F. [tex]$H_0: \bar{x}=110.2, \quad H_A: \bar{x} \neq 110.2$[/tex]

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.