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The image depicts a normal distribution curve, and the problem asks to find the \( z \)-value that corresponds to a given area. The shaded area under the curve is 0.3012, which represents the cumulative probability to the left of a certain \( z \)-score. The graph shows that this area is located on the left side of the mean, so the \( z \)-value will be negative.

To solve for the \( z \)-value:
1. Use the cumulative probability (0.3012).
2. Look up the value in a standard normal distribution table (or use a graphing calculator or software like a normal distribution calculator) to find the corresponding \( z \)-value.

By using a standard normal distribution table or calculator, we find that the \( z \)-value corresponding to a cumulative probability of 0.3012 is approximately **-0.5244**.

Would you like more details on how this was calculated, or do you have any questions?

Here are 5 related questions to expand on this concept:
1. What is the significance of a negative \( z \)-score in a normal distribution?
2. How can you calculate \( z \)-scores for different cumulative probabilities?
3. Why is the area under the curve important in statistics?
4. How is the \( z \)-score related to standard deviations from the mean?
5. How can you use technology (like a graphing calculator) to quickly find \( z \)-scores?

**Tip**: You can always use inverse normal functions in calculators or statistical software to find \( z \)-scores from a given cumulative probability.

Find the z-value that corresponds to the given area of 0.3012 under the normal distribution curve.

Learn how to find the z-value corresponding to a cumulative probability of 0.3012 in a standard normal distribution. This problem uses concepts like the z-score, cumulative probability, and inverse normal distribution functions. Suitable for students in Grades 10-12 or introductory college-level statistics.

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