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Let's analyze the math content from your AP Statistics Quiz.

### Problem Breakdown:
1. **The context:**
   - A study was conducted to assess whether the "accepted" normal body temperature (98.6°F) is accurate.
   - 130 healthy individuals' body temperatures were collected, summarized in the table and dotplot provided.
   
   The table shows:
   - Mean = 36.805°C
   - SE Mean (Standard Error) = 0.036
   - Standard Deviation = 0.408
   - Minimum = 35.700°C
   - Quartiles: Q1 = 36.550°C, Median = 36.850°C, Q3 = 37.050°C
   - Maximum = 38.200°C

### (a) Interpretation of Mr. Withey's temperature at the 20th percentile:
   - The 20th percentile means Mr. Withey's temperature is higher than 20% of the temperatures in the distribution. 
   - This means his temperature is lower than most, but still falls within the lower range of the data.

### (b) What was Mr. Withey's temperature?
   - Based on the question's context, you likely need to estimate the temperature corresponding to the 20th percentile. Since the temperature data is summarized with quartiles, and the 20th percentile is between the minimum (35.7°C) and Q1 (36.55°C), Mr. Withey's temperature would be in this range, closer to Q1. We could estimate his temperature to be approximately **36.3°C**.

### (c) Convert the mean temperature to Fahrenheit:
   The formula given is:
   \[
   ^\circ F = \left(\frac{9}{5}\right) \cdot ^\circ C + 32
   \]
   To convert the mean temperature of 36.805°C:
   \[
   ^\circ F = \left(\frac{9}{5}\right) \cdot 36.805 + 32
   \]

   Let me calculate that for you.The mean body temperature, when converted from 36.805°C to Fahrenheit, is approximately **98.25°F**.

### Interpretation of Part (c):
Since the "accepted" normal body temperature is 98.6°F, the calculated mean of 98.25°F suggests that the actual average body temperature of the 130 healthy individuals is slightly lower than the historically accepted value.

Would you like further details on any of these steps? 

Here are five questions that could expand your understanding:
1. What is the significance of the standard deviation in the context of this data?
2. How do we interpret the interquartile range (IQR) in this temperature distribution?
3. How could we estimate Mr. Withey’s exact temperature more accurately?
4. What factors could lead to the discrepancy between the mean temperature and the accepted value of 98.6°F?
5. How does the dotplot reflect the distribution's skewness or symmetry?

**Tip:** The percentile rank indicates the percentage of data points below a particular value, which helps in understanding where that value stands in a distribution.

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.
(a) Mr. Withey's temperature is at the 20th percentile of the distribution. Interpret this value in context.
(b) What was Mr. Withey's temperature?
(c) Find the mean temperature in degrees Fahrenheit. What does this suggest about the value of the 'normal' temperature?

Solve an AP Statistics problem involving the analysis of body temperature data. Learn how to interpret the 20th percentile in context, estimate a temperature, and convert the mean temperature from Celsius to Fahrenheit. Suitable for high school students in Grades 11-12.

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