The distribution of body temperature for healthy adults is approximately normal with mean 98.2°F and standard deviation 0.7°F. Based on the description of the distribution, which of the following statements CANNOT be supported?

To determine which statements cannot be supported, let's analyze each possible statement using the properties of a normal distribution.

Given:
- Mean (\(\mu\)) = 98.2°F
- Standard Deviation (\(\sigma\)) = 0.7°F
- The distribution is approximately normal.

### Key Points about Normal Distribution:
1. **68-95-99.7 Rule** (Empirical Rule):
   - 68% of the data falls within 1 standard deviation (\(\mu \pm 1\sigma\)) of the mean.
   - 95% of the data falls within 2 standard deviations (\(\mu \pm 2\sigma\)) of the mean.
   - 99.7% of the data falls within 3 standard deviations (\(\mu \pm 3\sigma\)) of the mean.

2. **Properties**:
   - About 50% of the data is below the mean, and about 50% is above the mean.
   - Almost all values lie within three standard deviations from the mean.

### Possible Statements Analysis:
Let's consider some example statements and analyze whether they can be supported or not:

1. **Statement: About 68% of healthy adults have body temperatures between 97.5°F and 98.9°F.**

   - Calculation: \(98.2 - 0.7 = 97.5\), \(98.2 + 0.7 = 98.9\)
   - This statement covers one standard deviation (\(\mu \pm 1\sigma\)) from the mean. According to the Empirical Rule, approximately 68% of the data falls within one standard deviation.
   - **Supported.**

2. **Statement: About 95% of healthy adults have body temperatures between 96.8°F and 99.6°F.**

   - Calculation: \(98.2 - 1.4 = 96.8\), \(98.2 + 1.4 = 99.6\)
   - This statement covers two standard deviations (\(\mu \pm 2\sigma\)) from the mean. According to the Empirical Rule, approximately 95% of the data falls within two standard deviations.
   - **Supported.**

3. **Statement: Less than 0.3% of healthy adults have body temperatures above 100.3°F.**

   - Calculation: \(98.2 + 3 \times 0.7 = 100.3\)
   - This statement is within three standard deviations (\(\mu + 3\sigma\)) from the mean. According to the Empirical Rule, only about 0.3% of the data lies beyond three standard deviations.
   - **Supported.**

4. **Statement: About 2.5% of healthy adults have body temperatures below 97.1°F.**

   - Calculation: \(98.2 - 1.5 \times 0.7 = 97.1\)
   - This is approximately 1.5 standard deviations below the mean. Roughly 2.5% of the data falls beyond this point (below \(\mu - 1.96\sigma\)).
   - **Supported.**

5. **Statement: Approximately 75% of healthy adults have body temperatures above 98.9°F.**

   - Calculation: \(98.9°F\) is 1 standard deviation above the mean (\(\mu + \sigma\)).
   - By the properties of the normal distribution, only about 16% of the data lies above one standard deviation from the mean, not 75%.
   - **Cannot be supported.**

### Conclusion:
- The statement that **"Approximately 75% of healthy adults have body temperatures above 98.9°F"** **cannot be supported** because, in a normal distribution, only about 16% of data is above one standard deviation from the mean.

Would you like more details or have any questions about this analysis?

### Related Questions:
1. What is the probability of a healthy adult having a body temperature above 99°F?
2. How do you calculate the z-score for a given body temperature?
3. What is the 90th percentile of body temperatures for healthy adults?
4. How would the distribution change if the standard deviation were smaller?
5. Can the mean body temperature vary across different populations?

### Tip:
When analyzing normal distributions, always refer to the Empirical Rule (68-95-99.7) to quickly estimate probabilities and check the plausibility of statements.

Explore how the normal distribution of body temperatures for healthy adults can be used to analyze statistical claims. Learn about the Empirical Rule, z-scores, and probabilities for data falling within specific ranges of the mean. Suitable for students in Grades 10-12.

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