A data set includes 103 body temperatures of healthy adult humans having a mean of 98.9 F and a standard deviation of 0.65 F. Construct a 99% confidence level estimate of the mean body temperature of all healthy humans. What does the sample suggest about the use of 98.6 F as the mean body temperature ? Find the confidence interval estimate of the population mean

To construct a 99% confidence interval for the mean body temperature of healthy adult humans, we will use the formula for a confidence interval based on the sample mean:

\[
\text{Confidence Interval} = \bar{x} \pm z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right)
\]

Where:
- \(\bar{x}\) is the sample mean,
- \(z_{\alpha/2}\) is the z-value corresponding to the confidence level,
- \(\sigma\) is the standard deviation of the sample,
- \(n\) is the sample size.

### Given Data:
- Sample size \(n = 103\),
- Sample mean \(\bar{x} = 98.9^\circ F\),
- Sample standard deviation \(\sigma = 0.65^\circ F\),
- Confidence level = 99%.

#### Step 1: Find the \(z_{\alpha/2}\) value for a 99% confidence level.
For a 99% confidence level, the \(z_{\alpha/2}\) value is approximately 2.576 (from standard normal distribution tables).

#### Step 2: Calculate the margin of error (E).
\[
E = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} = 2.576 \times \frac{0.65}{\sqrt{103}}
\]

Let me calculate this.The margin of error is approximately \( 0.165^\circ F \).

#### Step 3: Calculate the confidence interval.
The confidence interval is given by:

\[
\text{Confidence Interval} = \bar{x} \pm E = 98.9 \pm 0.165
\]

This results in the following interval:

\[
(98.735^\circ F, 99.065^\circ F)
\]

### Interpretation:
The 99% confidence interval for the mean body temperature of healthy adults is from \( 98.735^\circ F \) to \( 99.065^\circ F \).

### Conclusion on 98.6°F as the mean body temperature:
Since the value of \( 98.6^\circ F \) is not within the confidence interval, this sample suggests that \( 98.6^\circ F \) may not be an accurate estimate of the mean body temperature for healthy adults. The sample mean is higher, indicating that the commonly accepted value might be too low.

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Do you want further details or have any questions about this? Here are some related questions for practice:

1. What is the 95% confidence interval for the same data set?
2. How does increasing the sample size affect the margin of error?
3. If the confidence level were reduced to 90%, what would the interval be?
4. How would the interval change if the sample mean was lower?
5. Why is a 99% confidence interval wider than a 95% confidence interval?

**Tip:** Confidence intervals provide a range of values where the true population parameter is likely to lie, but the wider the interval, the more uncertainty about the exact value.

A data set includes 103 body temperatures of healthy adult humans having a mean of 98.9 F and a standard deviation of 0.65 F. Construct a 99% confidence level estimate of the mean body temperature of all healthy humans. What does the sample suggest about the use of 98.6 F as the mean body temperature? Find the confidence interval estimate of the population mean.

This problem constructs a 99% confidence interval for the mean body temperature of healthy humans using a sample size of 103, a mean of 98.9°F, and a standard deviation of 0.65°F. It challenges the traditional assumption of 98.6°F as the average body temperature.

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