https://file.aiquickdraw.com/mathbot/file/1727392110291xef6dhvv.jpg

The image you uploaded contains a multiple-choice statistics question from an AP Statistics examination practice. The question appears to involve calculating values related to a normal distribution.

Here is a breakdown of what I can see clearly:

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### Question:
1. The mean is 30 with a standard deviation of 5, and the 97.5th percentile has a value of 40. Which of the following is the best estimate for the mean and standard deviation of the variable?

(A) Mean = 32.5; Standard deviation = 5  
(B) Mean = 32.5; Standard deviation = 7.5  
(C) Mean = 30; Standard deviation = 5  
(D) Mean = 30; Standard deviation = 10

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It seems to be testing your ability to understand and estimate parameters of a normal distribution. The 97.5th percentile is key here, and from your diagram in the margin, it looks like you might have already drawn part of a normal distribution to visualize it.

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### Solving:

The problem involves a **normal distribution**. The 97.5th percentile in a normal distribution corresponds to approximately **two standard deviations above the mean**.

1. Given the values:
   - **Mean = 30**
   - **Standard deviation = 5**
   - **97.5th percentile value = 40**
   
   If the standard deviation is 5, then two standard deviations above the mean (30) would be:
   
   \( 30 + 2 \times 5 = 40 \)

This matches the information given, so the correct answer is **(C) Mean = 30, Standard deviation = 5**.

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Let me know if you need further clarification, details, or any additional help with this!

### Additional Practice Questions:

1. What does the 68-95-99.7 rule tell us about normal distributions?
2. How would you calculate the z-score of a value in a normal distribution?
3. What percentile corresponds to one standard deviation above the mean in a normal distribution?
4. How do you calculate variance from standard deviation?
5. Can you explain the difference between a population standard deviation and a sample standard deviation?

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### Tip: 
Always visualize normal distribution problems with a sketch and use the z-score table for precise percentile calculations.

The mean is 30 with a standard deviation of 5, and the 97.5th percentile has a value of 40. Which of the following is the best estimate for the mean and standard deviation of the variable?

(A) Mean = 32.5; Standard deviation = 5
(B) Mean = 32.5; Standard deviation = 7.5
(C) Mean = 30; Standard deviation = 5
(D) Mean = 30; Standard deviation = 10

Solve a normal distribution problem where the mean is 30 and standard deviation is 5, with the 97.5th percentile at 40. Learn how to estimate parameters and apply the empirical rule.

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