To construct a 99% confidence interval for the mean body temperature of healthy adults, we'll use the following formula:

$\text{Confidence Interval} = \bar{x} \pm z \left(\frac{\sigma}{\sqrt{n}}\right)$

where:

• $\bar{x}$ = sample mean = 98.2°F
• $\sigma$ = sample standard deviation = 0.64°F
• $n$ = sample size = 108
• $z$ is the z-score corresponding to a 99% confidence level.

Step 1: Determine the z-score for a 99% Confidence Level

For a 99% confidence interval, the z-score is approximately 2.576.

Step 2: Calculate the Margin of Error (ME)

$\text{Margin of Error (ME)} = z \times \left(\frac{\sigma}{\sqrt{n}}\right)$

Substitute the values:

$\text{ME} = 2.576 \times \left(\frac{0.64}{\sqrt{108}}\right)$

Let's calculate that:

1. Calculate $\sqrt{108} \approx 10.3923$.
2. Then, $\frac{0.64}{10.3923} \approx 0.0616$.
3. Finally, $2.576 \times 0.0616 \approx 0.1587$.

So, the margin of error is approximately 0.159°F.

Step 3: Construct the Confidence Interval

$\text{Confidence Interval} = \bar{x} \pm \text{ME}$ $= 98.2 \pm 0.159$

This gives us:

$98.2 - 0.159 = 98.041 \quad \text{and} \quad 98.2 + 0.159 = 98.359$

Confidence Interval

Thus, the 99% confidence interval for the mean body temperature is:

$98.041^\circ \text{F} < \mu < 98.359^\circ \text{F}$

Interpretation Regarding 98.6°F as a Mean Body Temperature

Since 98.6°F is outside this confidence interval, the sample suggests that the true mean body temperature of healthy humans might be lower than the traditionally accepted value of 98.6°F.

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Would you like further details on any part of this calculation?

Here are some related questions to explore:

1. How does sample size impact the width of a confidence interval?
2. Why is a z-score used instead of a t-score in this example?
3. How would the confidence interval change if the confidence level were 95% instead of 99%?
4. What are the implications of finding a different mean for health-related norms?
5. How does standard deviation affect the margin of error in confidence intervals?

Tip: For larger samples (n > 30), the z-score is typically used for confidence intervals, even if the population standard deviation is unknown.