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.

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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.

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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.

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Would you like more details or further clarifications?

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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!