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.