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The problem you're working on involves a normally distributed set of thermometer readings with a mean of 0°C and a standard deviation of 1°C. The goal is to find \( P_{87} \), the 87th percentile of this distribution, which separates the bottom 87% of data from the top 13%.

From the image, it looks like you have already identified the correct graph (choice C), which highlights the region of the normal distribution to the left of \( P_{87} \), indicating that the area under the curve to the left of this point is 87%.

### To find the temperature corresponding to \( P_{87} \):
1. First, find the z-score that corresponds to the 87th percentile. Using z-tables or statistical software, the z-score for the 87th percentile is approximately:
   \[
   z = 1.13
   \]
2. Next, use the formula for converting a z-score to a temperature in this normally distributed set:
   \[
   T = \mu + z \cdot \sigma
   \]
   where \( \mu = 0 \) (mean) and \( \sigma = 1 \) (standard deviation).

   Substituting the values:
   \[
   T = 0 + (1.13 \cdot 1) = 1.13^\circ C
   \]

Therefore, the temperature corresponding to \( P_{87} \) is approximately \( 1.13^\circ C \).

Would you like a more detailed explanation or have any questions? Here are some related questions to expand on this:

1. What is the significance of the z-score in a normal distribution?
2. How would you calculate the z-score for other percentiles like the 95th or 50th?
3. Why is the normal distribution widely used in statistics?
4. How do you calculate probabilities in normal distributions for values not directly at the mean?
5. What are the properties of a standard normal distribution?

**Tip:** When working with percentiles in a normal distribution, always remember that the area under the curve to the left of a given point equals the percentile value.

Assume that the readings on the thermometers are normally distributed with a mean of 0°C and standard deviation of 1.00°C. A thermometer is randomly selected and tested. Draw a sketch and find the temperature reading corresponding to P87, the 87th percentile. This is the temperature reading separating the bottom 87% from the top 13%.

Learn how to find the temperature corresponding to the 87th percentile in a normally distributed set of thermometer readings, given a mean of 0°C and a standard deviation of 1°C. This problem explores the use of z-scores and normal distribution properties.

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