the claim is that for 12 AM body temperatures, the mean is less than 98.6 F. the sample size is n = 6 and the test statisitic is t = -2.255. find the P-value for the hypothesis test described based on the result, what is the final conclusion? use a significance level of a = 0.05. State the null and alternative hypotheses.

We are conducting a hypothesis test for the mean body temperature. Here’s a step-by-step breakdown of the process:

### Step 1: State the Null and Alternative Hypotheses
The claim is that the mean body temperature at 12 AM is less than 98.6°F. Therefore, the hypotheses are:

- **Null Hypothesis (\(H_0\))**: The mean body temperature is 98.6°F or higher.  
  \[ H_0: \mu = 98.6 \text{°F} \]
- **Alternative Hypothesis (\(H_a\))**: The mean body temperature is less than 98.6°F.  
  \[ H_a: \mu < 98.6 \text{°F} \]

This is a **left-tailed test** since we are testing if the mean is **less than** 98.6°F.

### Step 2: Determine the Test Statistic
The test statistic given is:
\[
t = -2.255
\]

The sample size is \(n = 6\). Degrees of freedom (\(df\)) for the t-distribution is calculated as:
\[
df = n - 1 = 6 - 1 = 5
\]

### Step 3: Find the P-value
To find the P-value, we need to look up the t-distribution with 5 degrees of freedom and the test statistic \( t = -2.255 \). Since this is a left-tailed test, the P-value is the area to the left of \( t = -2.255 \) in the t-distribution.

Using a t-distribution table or calculator, we can find the P-value for \( t = -2.255 \) and \( df = 5 \).

### Step 4: Conclusion
Once we have the P-value, we can compare it to the significance level \( \alpha = 0.05 \):

- If the **P-value is less than 0.05**, we reject the null hypothesis \(H_0\).
- If the **P-value is greater than 0.05**, we fail to reject the null hypothesis \(H_0\).

Let me calculate the P-value for this test next.The P-value is approximately 0.0369.

### Step 5: Conclusion
- Since the P-value (0.0369) is **less than** the significance level \( \alpha = 0.05 \), we **reject the null hypothesis**.

### Final Conclusion:
There is enough evidence to support the claim that the mean body temperature at 12 AM is less than 98.6°F.

---

Would you like further details or have any questions? Here are some additional questions to deepen your understanding:

1. What would happen if the significance level were set to \( \alpha = 0.01 \)?
2. How do changes in sample size affect the P-value in t-tests?
3. Why is the test considered a left-tailed test in this context?
4. What assumptions are necessary when performing a t-test like this one?
5. How would the conclusion change if the test statistic were positive instead of negative?

**Tip:** In hypothesis testing, always check if your test is one-tailed or two-tailed, as this affects how you calculate the P-value.

The claim is that for 12 AM body temperatures, the mean is less than 98.6 F. The sample size is n = 6 and the test statistic is t = -2.255. Find the P-value for the hypothesis test described. Based on the result, what is the final conclusion? Use a significance level of a = 0.05. State the null and alternative hypotheses.

Solve a hypothesis test for the claim that the mean body temperature at 12 AM is less than 98.6°F. Given a sample size of n = 6 and test statistic t = -2.255, find the P-value and compare it to a significance level of 0.05 to draw a conclusion. Includes step-by-step calculations and explanation of t-distribution.

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