https://file.aiquickdraw.com/mathbot/file/1728863421937pezlizl7.jpg

This problem involves using the normal distribution to calculate Amina's SAT Math score, given that her score corresponds to the 61st percentile. Here's how we solve it:

1. **Mean (μ) and standard deviation (σ)**:
   - The mean SAT Math score: \( \mu = 522 \)
   - The standard deviation: \( \sigma = 214.5 \)

2. **Percentile to z-score**:
   - The 61st percentile corresponds to a z-score of approximately 0.305 (which matches the z-score given in your image).

3. **Finding Amina’s SAT Math score**:
   We use the formula for converting a z-score to a raw score:
   \[
   X = \mu + z \cdot \sigma
   \]
   Plugging in the values:
   \[
   X = 522 + 0.305 \times 214.5
   \]
   \[
   X \approx 522 + 65.4225 \approx 587.42
   \]
   Rounded to the nearest whole number, Amina’s SAT Math score is **588**, which matches the answer given in the image.

### Summary:
- The z-score used is **0.305**.
- Amina's SAT Math score is **588** points.

Let me know if you'd like further clarification! Here are five related questions to deepen your understanding:

1. What is the z-score for a score that corresponds to the 90th percentile?
2. How would Amina’s score change if the standard deviation were 250 instead of 214.5?
3. What score corresponds to the 50th percentile (median) for this distribution?
4. How is the z-score related to standard deviations from the mean?
5. What percentile would correspond to a score of 650?

**Tip**: When working with percentiles, remember that the z-score represents how many standard deviations a data point is from the mean. This is crucial for converting between z-scores and raw scores.

In 1986, student scores on the SAT Math Test were normally distributed, with a mean of 522 and a standard deviation of 214.5. Amina was informed that she scored in the 61st percentile. What was Amina's score of the SAT Math Test?

Learn how to calculate a SAT Math score from the 61st percentile using the normal distribution, with a mean of 522 and standard deviation of 214.5. This tutorial covers essential concepts in statistics, including z-scores and percentile calculations. Suitable for high school students.

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