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 ?

To construct a 99% confidence interval estimate of the mean body temperature of all healthy humans based on the sample provided, we'll use the formula for a confidence interval for the mean:

\[
\text{CI} = \bar{x} \pm z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
\]

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

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

### Step 1: Find \(z_{\alpha/2}\)
For a 99% confidence level, the critical value \(z_{\alpha/2}\) corresponds to the area in both tails being \(1 - 0.99 = 0.01\), so \(z_{\alpha/2}\) covers 0.005 in each tail. Using the standard normal distribution table or calculator, we get:
\[
z_{\alpha/2} = 2.576
\]

### Step 2: Calculate the margin of error (E)
The margin of error is given by:
\[
E = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
\]
Substituting the values:
\[
E = 2.576 \times \frac{0.65}{\sqrt{103}} \approx 2.576 \times 0.0641 \approx 0.165
\]

### Step 3: Construct the confidence interval
Now, we calculate the confidence interval by adding and subtracting the margin of error from the sample mean:
\[
\text{CI} = 98.9 \pm 0.165
\]
Thus, the 99% confidence interval is:
\[
(98.735^\circ F, 99.065^\circ F)
\]

### Step 4: Interpret the result
This confidence interval suggests that we are 99% confident that the true mean body temperature of healthy humans lies between \(98.735^\circ F\) and \(99.065^\circ F\).

### Implications for the use of \(98.6^\circ F\)
Since the value \(98.6^\circ F\) (the traditionally accepted average body temperature) is **not** within this confidence interval, the data suggests that the mean body temperature of healthy humans is likely higher than \(98.6^\circ F\). Therefore, using \(98.6^\circ F\) as the average may no longer be accurate based on this sample.

Would you like to explore more details or any other concepts related to this topic?

---

Here are five related questions you might find interesting:
1. How is the critical value \(z_{\alpha/2}\) determined for other confidence levels?
2. What would the confidence interval be at a 95% confidence level?
3. How does increasing the sample size affect the confidence interval width?
4. What are the differences between confidence intervals for the population mean when the population standard deviation is unknown?
5. How can we interpret confidence intervals in the context of hypothesis testing?

**Tip:** Remember that increasing the confidence level makes the confidence interval wider, but also increases certainty in capturing the true population mean.

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?

This guide explains how to construct a 99% confidence interval for the mean body temperature of healthy adults using a sample of 103. It also explores the implications of using 98.6°F as the traditional average temperature.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation