https://file.aiquickdraw.com/mathbot/file/1729097515986sq61b5th.jpg

The image shows a normal distribution curve with a shaded area of 0.9456 to the left of the value \( z \). The goal is to find the corresponding \( z \)-score that leaves an area of 0.9456 to the left of it.

To find the \( z \)-value:

1. The cumulative area of 0.9456 represents the cumulative probability up to \( z \) under the standard normal curve.
2. Using a standard normal distribution table or a graphing calculator, find the \( z \)-score corresponding to an area of 0.9456.

Using standard tables or a calculator, the \( z \)-value corresponding to an area of 0.9456 is approximately:

\[
z \approx 1.60
\]

Thus, \( z = 1.60 \).

Would you like more details or have any questions about this?

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Here are 5 questions for further exploration:
1. How does the cumulative distribution function (CDF) help in finding the \( z \)-score?
2. What is the meaning of a standard normal distribution?
3. How would you find the \( z \)-value if the area given was less than 0.5?
4. Can the \( z \)-score be negative? If yes, when does that happen?
5. How can the standard normal distribution be applied in real-world problems?

**Tip:** Always use a standard normal table or calculator for quick reference to cumulative probabilities in normal distributions.

Find the z-value that corresponds to the given area in the figure below.

Discover how to find the z-value for a cumulative area of 0.9456 under the standard normal curve. This problem introduces key statistical concepts like z-scores and cumulative probability, providing a step-by-step solution. Perfect for high school students and those studying introductory statistics.

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