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.

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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.