Find the indicated​ z-scores shown in the graph.

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z=?z=?0x0.02810.0281

A normal curve is over a horizontal x-axis and is centered on 0. Vertical line segments extend from the curve to the horizontal axis at two points labeled z = ? each. The areas under the curve and to the left of the left vertical line segment and to the right of the right vertical line segment are shaded and labeled 0.0281 each.

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Part 1

The​ z-scores are

enter your response here.

​(Use a comma to separate answers as needed. Round to two decimal places as​ needed.)

The problem asks for the \( z \)-scores corresponding to a total area of 0.0281 in both tails of a normal distribution. Given the symmetry of the normal curve, this area is divided equally between the left and right tails, with each shaded region representing an area of 0.0281.

To find the \( z \)-scores:

1. The total area under the curve to the left of the left tail's \( z \)-score is \( 0.0281 \). This is the area in the lower tail. The cumulative probability corresponding to the left \( z \)-score is \( 0.0281 \).
2. The area to the right of the right tail's \( z \)-score is also \( 0.0281 \). The cumulative probability for this is \( 1 - 0.0281 = 0.9719 \), because the total area under the curve is 1.

Using a standard normal distribution table (or a calculator), we can find the \( z \)-scores that correspond to these cumulative probabilities:

- For a cumulative probability of \( 0.0281 \), the \( z \)-score is approximately \( -1.91 \).
- For a cumulative probability of \( 0.9719 \), the \( z \)-score is approximately \( 1.91 \).

Thus, the \( z \)-scores are approximately \( -1.91 \) and \( 1.91 \).

**Answer:** The \( z \)-scores are \( -1.91 \) and \( 1.91 \).

Would you like more details or have any further questions?

Here are 5 related questions for further exploration:
1. How do you calculate a \( z \)-score from a given probability?
2. What is the relationship between cumulative probability and the \( z \)-score?
3. How would the \( z \)-score change if the shaded area were 0.05 instead of 0.0281?
4. Can you explain the concept of tail probability in a normal distribution?
5. What is the empirical rule, and how does it relate to \( z \)-scores?

**Tip:** When looking up \( z \)-scores for specific probabilities, remember that negative \( z \)-scores correspond to areas on the left of the mean, while positive ones are on the right.

Find the indicated z-scores shown in the graph. A normal curve is over a horizontal x-axis and is centered on 0. Vertical line segments extend from the curve to the horizontal axis at two points labeled z = ?. The areas under the curve and to the left of the left vertical line segment and to the right of the right vertical line segment are shaded and labeled 0.0281 each.

Learn how to find the z-scores corresponding to shaded areas in a standard normal distribution. This problem involves calculating z-scores from cumulative probabilities and understanding the properties and symmetry of the normal curve. Suitable for students in Grades 10-12.

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