The minimum temperature for requiring further medical tests should be

°F if we want only 5.0% of healthy people

to exceed it.

(Round to two decimal places as needed.)

To determine the minimum temperature that only 5.0% of healthy people exceed, we need to assume that body temperatures follow a normal distribution. Let’s denote the mean and standard deviation of healthy body temperatures as \( \mu \) and \( \sigma \), respectively. Then we are looking for the temperature, \( T \), such that the proportion of healthy individuals exceeding this temperature is 5%.

### Steps:
1. For a normal distribution, the threshold that leaves 5% of the population in the upper tail corresponds to a \( z \)-score of approximately \( z = 1.645 \) (since 95% lies below it, and 5% is above it).
   
2. The relationship between the temperature \( T \), the mean \( \mu \), the standard deviation \( \sigma \), and the \( z \)-score is:
   \[
   T = \mu + z \cdot \sigma
   \]
   
3. We need the values of \( \mu \) and \( \sigma \) for healthy body temperatures to solve this equation. A typical estimate for normal body temperature is \( \mu = 98.6^\circ F \) and the standard deviation \( \sigma = 0.7^\circ F \) (although this can vary).

Now, using these values:
\[
T = 98.6 + 1.645 \times 0.7
\]
\[
T = 98.6 + 1.1515 = 99.75^\circ F
\]

So, the minimum temperature for requiring further medical tests should be approximately **99.75°F**.

### Do you want more details or have any questions?

Here are five related questions:
1. What is the z-score and how is it used in this context?
2. What if the mean body temperature or the standard deviation changes?
3. Can you explain how the normal distribution relates to real-world measurements like temperature?
4. How does the 5% threshold apply to other medical screenings?
5. What would be the corresponding Celsius temperature for this threshold?

**Tip**: When using normal distribution, the area under the curve helps you understand the proportion of data within certain ranges of standard deviation.

The minimum temperature for requiring further medical tests should be _____°F if we want only 5.0% of healthy people to exceed it.

Learn how to calculate the minimum temperature that only 5% of healthy people exceed using the z-score and normal distribution. This problem covers key concepts like normal distribution, percentiles, and the z-score formula, making it suitable for Grades 10-12.

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