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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.)

Choose the correct set of hypotheses:

A. [tex]H_0: \mu = 110.2, \quad H_A: \mu < 110.2[/tex]

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

C. [tex]H_0: \mu = 115, \quad H_A: \mu > 115[/tex]

D. [tex]H_0: \bar{x} = 115, \quad H_A: \bar{x} > 115[/tex]

E. [tex]H_0: \mu = 115, \quad H_A: \mu < 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 need to select the correct null hypothesis ([tex]\(H_0\)[/tex]) and alternative hypothesis ([tex]\(H_A\)[/tex]) based on the given information about the sheep dog's heart rate.

Here's how to approach it step-by-step:

1. Understand the Context: The farmer believes his sheep dog's heart rate is slower than the average for her breed. The average resting heart rate for this type of sheep dog is 115 beats per minute.

2. Identify the Given Data:
- Average (expected) heart rate for the breed ([tex]\(\mu\)[/tex]): 115 bpm
- Sample mean heart rate for the farmer's dog ([tex]\(\bar{x}\)[/tex]): 110.2 bpm

3. Formulate Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): This is the statement we want to test against. Generally, it reflects the status quo or no change condition. In this context, [tex]\(H_0\)[/tex] should reflect the typical mean heart rate for the breed, which is [tex]\(\mu = 115\)[/tex].
- Alternative Hypothesis ([tex]\(H_A\)[/tex]): This is what the farmer suspects, that the heart rate is slower than normal (less than 115 bpm). Hence, [tex]\(H_A\)[/tex] should be [tex]\(\mu < 115\)[/tex].

4. Select the Correct Option:
- We need to pick the option that matches our null and alternative hypotheses.
- Review the options provided:
- A. [tex]\(H_0: \mu = 110.2, \quad H_A: \mu < 110.2\)[/tex]
- B. [tex]\(H_0: \mu = 115, \quad H_A: \mu \neq 115\)[/tex]
- C. [tex]\(H_0: \mu = 115, \quad H_A: \mu > 115\)[/tex]
- D. [tex]\(H_0: \bar{x} = 115, \quad H_A: \bar{x} > 115\)[/tex]
- E. [tex]\(H_0: \mu = 115, \quad H_A: \mu < 115\)[/tex]
- F. [tex]\(H_0: \bar{x} = 110.2, \quad H_A: \bar{x} \neq 110.2\)[/tex]

- The correct choice that fits our hypotheses ([tex]\(H_0: \mu = 115\)[/tex] and [tex]\(H_A: \mu < 115\)[/tex]) is option E.

Thus, the correct hypotheses for this scenario are:

- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 115\)[/tex]
- Alternative Hypothesis ([tex]\(H_A\)[/tex]): [tex]\(\mu < 115\)[/tex]

This means we are testing if the heart rate of the farmer's sheep dog is indeed slower than the average for its breed.

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