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.