For more than a century, doctors have been telling patients that a normal body temperature is 98.6 °F. This value dates back to a study done by Carl Wunderlich in the mid-1800s. More recently, researchers conducted a study to determine whether the “accepted” value for normal body temperature is accurate. They collected body temperatures (in degrees Celsius) from 130 healthy individuals. A dotplot and numerical summaries of the data are given below

Variable Count Mean SE StDev Minimum Q1 Median Q3 Maximum Temp 

130 36.805 0.036 0.408 35.700 36.550 36.850 37.050 38.200 

(a)Mr. Withey’s temperature is at the 20th percentile of the distribution. Interpret this value in context.

(b)what was Mr whitey's temperature?

(c) find the mean temperature in degrees faranheit using that F=(9/5)(C) + 32. What does this suggest about the value of the 'normal' body temperature?

Let's go through each part of the problem step by step.

### (a) Interpretation of the 20th Percentile
Mr. Withey’s temperature being at the 20th percentile means that 20% of the individuals in the study have a body temperature that is **less than or equal to** his temperature. In other words, Mr. Withey’s temperature is higher than or equal to the temperatures of 20% of the individuals, and lower than the temperatures of the remaining 80% of the individuals in the distribution.

### (b) Mr. Withey's Temperature
To determine Mr. Withey's temperature, we need to use the fact that he is at the 20th percentile. From the numerical summary provided:

- **Q1** (25th percentile) is 36.550°C.
- The 20th percentile is likely just below Q1.

Given the dataset's distribution and summary statistics, Mr. Withey’s temperature is slightly below 36.550°C. To find the exact value, we might interpolate between the minimum value and Q1. However, without the raw data, we cannot pinpoint the exact temperature, but it's likely close to 36.5°C.

### (c) Mean Temperature in Fahrenheit
The mean temperature in degrees Celsius is 36.805°C. To convert this to Fahrenheit, we use the formula:

\[
F = \left(\frac{9}{5}\right) \times C + 32
\]

Substituting the mean temperature in Celsius:

\[
F = \left(\frac{9}{5}\right) \times 36.805 + 32
\]

Let's calculate this value.

\[
F = \left(1.8\right) \times 36.805 + 32 = 66.249 + 32 = 98.249°F
\]

The mean temperature in Fahrenheit is approximately **98.25°F**.

### Interpretation
The mean temperature of 98.25°F suggests that the widely accepted "normal" body temperature of 98.6°F might be slightly higher than the average body temperature observed in the study. This suggests that the traditional value of 98.6°F may not accurately reflect the typical body temperature of healthy individuals today.

---

Do you have any questions or need further details? Here are some related questions for further exploration:

1. How is the 20th percentile calculated from raw data?
2. How would the standard deviation affect the interpretation of the "normal" body temperature?
3. What are some reasons why the mean body temperature might differ from Wunderlich's historical value?
4. How would you convert the Q1 and Q3 temperatures to Fahrenheit?
5. What could be the implications of relying on a possibly outdated "normal" body temperature in medical practice?

**Tip:** Percentiles are useful in comparing an individual's measurement to a distribution and understanding where they stand relative to the population.

Explore the interpretation of percentiles and average body temperature based on statistical analysis. Learn about the accuracy of traditional body temperature norms and their implications in medical practice. Includes calculations and insights derived from recent research.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation