Assume a significance level of $$ \alpha=0.05 $$ and use the given information to complete parts (a) and (b) below. Original claim: The mean pulse rate (in beats per minute) of a certain group of adult males is 76 bpm. The hypothesis test results in a P-value of 0.0697 . B. Reject $$ H_{0} $$ because the P-value is less than or equal to $$ \alpha $$. Fail to reject $$ H_{0} $$ because the P-value is greater than $$ \alpha $$. D. Fail to reject $$ H_{0} $$ because the P-value is less than or equal to $$ \alpha $$. b. Without using technical terms, state a final conclusion that addresses the original claim. Which of the following is the correct conclusion? A. There is sufficient evidence to warrant rejection of the claim that the mean pulse rate (in beats per minute) of the group of adult males is 76 bpm. B. The mean pulse rate (in beats per minute) of the group of adult males is not 76 bpm. C. The mean pulse rate (in beats per minute) of the group of adult males is 76 bpm. D. There is not sufficient evidence to warrant rejection of the claim that the mean pulse rate (in beats per minute) of the group of adult males is 76 bpm.

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To solve this problem, we need to analyze the hypothesis test results based on the provided P-value and significance level.

### Known Conditions:
- Original claim (null hypothesis, $$ H_0 $$): The mean pulse rate of a certain group of adult males is 76 bpm.
- Significance level ($$ \alpha $$): 0.05
- P-value: 0.0697

### Part (a):
We need to determine whether to reject or fail to reject the null hypothesis based on the P-value and significance level.

1. **Compare the P-value to $$ \alpha $$**:
   - P-value (0.0697) is greater than $$ \alpha $$ (0.05).

2. **Decision**:
   - Since the P-value is greater than $$ \alpha $$, we **fail to reject $$ H_0 $$**.

Thus, the correct choice for part (a) is:
- **Fail to reject $$ H_{0} $$ because the P-value is greater than $$ \alpha $$**.

### Part (b):
Now, we need to state a final conclusion that addresses the original claim.

1. **Interpretation of failing to reject $$ H_0 $$**:
   - Failing to reject the null hypothesis means that we do not have enough evidence to say that the mean pulse rate is different from 76 bpm.

2. **Conclusion**:
   - We need to choose the statement that reflects this interpretation.

The correct conclusion is:
- **D. There is not sufficient evidence to warrant rejection of the claim that the mean pulse rate (in beats per minute) of the group of adult males is 76 bpm.**

### Summary:
- Part (a): Fail to reject $$ H_{0} $$ because the P-value is greater than $$ \alpha $$.
- Part (b): D. There is not sufficient evidence to warrant rejection of the claim that the mean pulse rate (in beats per minute) of the group of adult males is 76 bpm.
- **Part (a):** Fail to reject $$ H_{0} $$ because the P-value (0.0697) is greater than $$ \alpha $$ (0.05). - **Part (b):** There is not sufficient evidence to warrant rejection of the claim that the mean pulse rate of the group of adult males is 76 bpm.

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