To solve these problems, we need to use the properties of the normal distribution. Specifically, we will use the z-score formula and the standard normal distribution table to find the corresponding probabilities.

Problem (a)

You are asked to determine the percentage of normal and healthy persons who would be considered to have a fever using a cutoff of 100.6°F.

Given:

• Mean ($\mu$) = 98.22°F
• Standard deviation ($\sigma$) = 0.62°F
• Cutoff for fever = 100.6°F

We need to find the percentage of people who have a temperature above 100.6°F, so we calculate the z-score:

$z = \frac{x - \mu}{\sigma}$ where:

• $x$ = 100.6°F (the cutoff temperature)
• $\mu$ = 98.22°F (mean)
• $\sigma$ = 0.62°F (standard deviation)

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

Using a standard normal distribution table, a z-score of 3.84 corresponds to a cumulative probability of 0.9999. This means that 99.99% of the population will have a temperature below 100.6°F, and the percentage of people considered to have a fever is:

$1 - 0.9999 = 0.0001 = 0.01\%$

So, 0.01% of normal and healthy persons would be considered to have a fever.

Interpretation:

This percentage is very small, which suggests that a cutoff of 100.6°F is quite high. So, the answer would be D: Yes, because there is a small probability that a normal and healthy person would be considered to have a fever.

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Problem (b)

Physicians want to select a minimum temperature for requiring further tests such that only 5% of healthy people exceed this temperature.

This means we need to find the temperature that corresponds to the top 5% of the normal distribution. The cumulative probability for the top 5% corresponds to a cumulative probability of 0.9500.

Using a standard normal distribution table or a calculator, we find that the z-score for 0.9500 is approximately 1.645.

We can now use the z-score formula to find the temperature ($x$):

$z = \frac{x - \mu}{\sigma}$

Rearrange to solve for $x$:

$x = z \cdot \sigma + \mu$

Substitute the known values:

$x = 1.645 \cdot 0.62 + 98.22$ $x \approx 1.0199 + 98.22 = 99.24°F$

So, the minimum temperature for requiring further medical tests should be approximately 99.24°F.

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Summary:

• (a) The percentage of normal and healthy persons considered to have a fever is 0.01%.
  • The answer is D: Yes, because there is a small probability that a normal and healthy person would be considered to have a fever.
• (b) The minimum temperature for requiring further medical tests should be 99.24°F.

Would you like further details on how to use z-scores, or do you have any other questions?

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Relative Questions:

1. How would the results change if the standard deviation of body temperature was higher?
2. What percentage of people have body temperatures between 98°F and 100.6°F?
3. What is the z-score for a body temperature of 99.5°F?
4. If we use a cutoff of 100.0°F instead, what percentage of healthy people would be considered to have a fever?
5. How can physicians balance false positives and false negatives when setting a fever threshold?

Tip:

When working with normal distributions, always make sure to use the z-score to translate between raw scores and probabilities.