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Part 1
Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.

z0.1762
0

A symmetric bell-shaped curve is plotted over a horizontal scale with two labeled coordinates. One coordinate is labeled "0" and is located at the center and peak of the curve. The other coordinate is labeled "z," and is to the left of 0. A vertical line extends from the scale to the curve at z. The area under the curve to the left of z is shaded and labeled “0.1762.”
Question content area bottom
Part 1
The indicated z score is

enter your response here.
​(Round to two decimal places as​ needed.)

To find the indicated z-score for a given cumulative area in a standard normal distribution, you can use the cumulative distribution function (CDF) for the standard normal distribution. The area to the left of the z-score \( z \) is given as 0.1762. 

To find the z-score corresponding to this cumulative area, we can use the inverse of the standard normal distribution, often denoted as \( \Phi^{-1}(p) \), where \( p \) is the cumulative area. 

Here's the step-by-step process:

1. Identify the cumulative area to the left of the z-score, which is 0.1762.
2. Use the inverse of the standard normal distribution (the z-table or a statistical calculator) to find the z-score corresponding to this cumulative area.

Using statistical tables or a calculator, we can find the z-score \( z \) such that \( \Phi(z) = 0.1762 \).

Using a standard normal distribution table or a statistical tool:

\[ z \approx \Phi^{-1}(0.1762) \]

Looking up this value in a z-table or using a statistical calculator:

\[ z \approx -0.93 \]

Therefore, the indicated z-score is approximately:

\[ z \approx -0.93 \]

If you need further details or have any questions, please let me know!

Here are some related questions you might find helpful:

1. What is the z-score corresponding to a cumulative area of 0.25?
2. How do you find the z-score if the area to the right is given instead of the left?
3. What is the area under the standard normal curve between \( z = -1 \) and \( z = 1 \)?
4. How do you interpret a z-score in a standard normal distribution?
5. How can you find the z-score for a given percentile?
6. What is the relationship between the z-score and standard deviations from the mean?
7. How do you use a z-table to find cumulative areas?
8. What is the cumulative area to the left of \( z = 0 \) in a standard normal distribution?

**Tip:** When working with z-scores, always remember that the standard normal distribution has a mean of 0 and a standard deviation of 1. This helps simplify calculations and interpretations.

Learn how to find the indicated z-score for a given cumulative area in a standard normal distribution. This problem covers the use of the cumulative distribution function (CDF) and the inverse of the standard normal distribution. Suitable for college-level statistics and mathematics students.

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