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In this problem, you are being asked to evaluate whether the pitcher Colón threw an easy pitch on purpose, based on a given normal curve calculation. The critical pitch speed in question is 85.7 mph, and you're expected to use a normal distribution to determine whether this speed was intentionally "easy" compared to his average pitches.

The given normal curve distribution has these variables:
- \(\mu\) (mean): This represents the average speed of Colón’s pitches.
- \(\sigma\) (standard deviation): This measures the spread or variability of the pitch speeds.

You are asked to calculate the cumulative probability of a pitch being 85.7 mph or lower, using a normal CDF (cumulative distribution function). Here’s a breakdown of how to approach this problem:

### Steps:
1. **Input the Mean and Standard Deviation**: You need to enter the mean \(\mu\) and standard deviation \(\sigma\) for Colón's pitch speeds. These might be available from past pitch data or typical pitching speeds in general.
   
2. **Set Upper Bound**: The speed of the particular pitch is 85.7 mph, which is the upper limit in this CDF calculation.
   
3. **Set Lower Bound**: This is set to -999 to ensure the full range of possible values below 85.7 mph is considered.

4. **Interpret the CDF Result**: Once the normal CDF is calculated, the result will give you the probability that a pitch thrown by Colón would be 85.7 mph or slower. If the probability is relatively high (close to 1), it suggests that this was not an outlier pitch, and he may not have thrown it purposely as an "easy" pitch. If the probability is very low, it could imply this pitch was indeed an outlier, which might support the idea that it was thrown intentionally slower.

If you want me to help with an assumed or given mean and standard deviation, I can calculate this for you. Let me know if you have any questions or need help with additional data for \(\mu\) and \(\sigma\)!

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Here are some follow-up questions:
1. What would be a reasonable mean and standard deviation for Colón's pitches?
2. How can the normal CDF function be used to interpret outliers?
3. What other statistical methods could be used to analyze pitch performance?
4. How does knowing the distribution of pitch speeds help assess a player's intent?
5. Would you expect a similar pitch distribution for different pitchers?

**Tip**: Always verify the context of your data—mean and standard deviation—in relation to real-world information when interpreting probabilities.

Do you think Colón purposefully threw an easy pitch? Justify your answer using a normal curve calculation.

Evaluate if a pitcher threw an easy pitch on purpose using a normal curve calculation. Learn to use the CDF of a normal distribution to interpret probabilities. Includes key concepts like mean, standard deviation, and statistical significance. Suitable for Grades 9-12.

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