Assume that human body temperatures are normally distributed with a mean of 98.22°F and a standard deviation of

0.62°F.

﻿﻿﻿A hospital uses 100.6°F as the lowest temperature considered to be a fever. What percentage of normal and healthy persons would be considered to have a fever? Does this percentage suggest that a cutoff of 100.6°F is appropriate?

﻿﻿﻿Physicians want to select a minimum temperature for requiring further medical tests. What should that temperature be, if we want only 5.0% of healthy people to exceed it? (Such a result is a false positive, meaning that the test result is positive, but the subject is not really sick.)

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a. The percentage of normal and healthy persons considered to have a fever is (Round to two decimal places as needed.)

%.

To solve the problem, we need to use the properties of the normal distribution and the z-score formula. The z-score formula is given by:

\[
z = \frac{x - \mu}{\sigma}
\]

where:
- \(x\) is the value we are examining (e.g., 100.6°F),
- \(\mu\) is the mean of the distribution (98.22°F),
- \(\sigma\) is the standard deviation (0.62°F).

### 1. **Finding the percentage of normal and healthy persons considered to have a fever:**

We are asked to find the percentage of healthy people who would be considered to have a fever, i.e., whose temperature exceeds 100.6°F. First, we calculate the z-score for 100.6°F:

\[
z = \frac{100.6 - 98.22}{0.62} = \frac{2.38}{0.62} \approx 3.84
\]

Now, we find the area to the right of \(z = 3.84\) in the standard normal distribution, which corresponds to the percentage of people with a temperature higher than 100.6°F. 

Looking up \(z = 3.84\) in a z-table or using a normal distribution calculator gives a probability of approximately 0.00006, or 0.006%. This means that only about **0.006% of normal and healthy people** would be considered to have a fever at 100.6°F.

### Conclusion on the appropriateness of 100.6°F cutoff:
Since such a tiny percentage of healthy individuals are classified as having a fever (only 0.006%), it suggests that a cutoff of 100.6°F is quite conservative. It would likely exclude many truly sick individuals, as very few healthy people would reach this temperature.

---

### 2. **Finding the temperature that only 5% of healthy people exceed (i.e., setting a 5% false positive rate):**

To find the temperature that only 5% of healthy people exceed, we need to find the z-score that corresponds to the upper 5% of the distribution. The z-score that leaves 5% in the upper tail is approximately \(z = 1.645\).

We can use the z-score formula to solve for the temperature \(x\):

\[
z = \frac{x - \mu}{\sigma}
\]

Substitute \(z = 1.645\), \(\mu = 98.22\), and \(\sigma = 0.62\):

\[
1.645 = \frac{x - 98.22}{0.62}
\]

Now, solve for \(x\):

\[
x - 98.22 = 1.645 \times 0.62 \approx 1.02
\]

\[
x = 98.22 + 1.02 = 99.24
\]

Thus, the temperature that only 5% of healthy people would exceed is approximately **99.24°F**. Setting this as the threshold would result in a 5% false positive rate.

---

### Summary of Answers:
1. The percentage of normal and healthy persons considered to have a fever at 100.6°F is **0.006%**.
2. To ensure that only 5% of healthy people exceed the threshold temperature, it should be set at **99.24°F**.

---

Would you like more details or further clarifications?

---

Here are 5 related questions:
1. How would the percentage change if the cutoff for fever was set lower, say at 100°F?
2. What is the significance of a z-score in a normal distribution?
3. How can this approach be used in other medical diagnostic tests involving false positives?
4. What are the risks of setting a temperature threshold too low or too high?
5. How does changing the standard deviation affect the percentage of false positives?

**Tip:** The z-score tells you how far a data point is from the mean in terms of standard deviations, making it a valuable tool for interpreting normal distributions!

Assume that human body temperatures are normally distributed with a mean of 98.22°F and a standard deviation of 0.62°F. A hospital uses 100.6°F as the lowest temperature considered to be a fever. What percentage of normal and healthy persons would be considered to have a fever? Does this percentage suggest that a cutoff of 100.6°F is appropriate? Physicians want to select a minimum temperature for requiring further medical tests. What should that temperature be, if we want only 5.0% of healthy people to exceed it?

Discover how to calculate the percentage of healthy individuals considered to have a fever at 100.6°F using the normal distribution. Learn about Z-scores, false positives, and determine the appropriate temperature threshold to reduce false positive rates to 5%. Ideal for students in Grades 10-12.

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