College

For each statement, select the correct null hypothesis, [tex]\( H_0 \)[/tex], and alternative hypothesis, [tex]\( H_A \)[/tex], in symbolic form.

(a) According to the Merck Veterinary Manual, the average resting heart rate for a certain type of sheep dog is 115 beats per minute (bpm). A Montana farmer notices his aging sheep dog 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 sheep dog over all occasions.)

A. [tex]\( H_0: \mu=115, \quad H_A: \mu\ \textgreater \ 115 \)[/tex]

B. [tex]\( H_0: \bar{x}=115, \quad H_A: \bar{x}\ \textgreater \ 115 \)[/tex]

C. [tex]\( H_0: \mu=115, \quad H_A: \mu\ \textless \ 115 \)[/tex]

D. [tex]\( H_0: \mu=110.2, \quad H_A: \mu\ \textless \ 110.2 \)[/tex]

E. [tex]\( H_0: \bar{x}=110.2, \quad H_A: \bar{x} \neq 110.2 \)[/tex]

F. [tex]\( H_0: \mu=115, \quad H_A: \mu \neq 115 \)[/tex]

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]