https://file.aiquickdraw.com/mathbot/file/1730073264161ds27dfe1.jpg

Let's solve each part of this question related to finding z-scores.

---

### Problem (a)
We are given two parts in (a) to find z-scores based on probabilities.

#### Part (i) \( P(Z < z) = 0.645 \)

To find \( z \) such that \( P(Z < z) = 0.645 \), we look up this probability in the standard normal distribution table or use statistical software.

From the z-table or software:
- \( z \approx 0.37 \) when \( P(Z < z) = 0.645 \).

Thus, the answer is:
- **\( z \approx 0.37 \)** (rounded to 2 decimal places).

---

#### Part (ii) \( P(Z > z) = 0.8 \)

For this part, we need to find \( z \) such that \( P(Z > z) = 0.8 \).

Since \( P(Z > z) = 0.8 \), then \( P(Z < z) = 1 - 0.8 = 0.2 \).

Using the z-table or statistical software:
- \( z \approx -0.84 \) when \( P(Z < z) = 0.2 \).

Therefore, the answer is:
- **\( z \approx -0.84 \)** (rounded to 2 decimal places).

---

### Problem (b)
In this part, we are asked to find the z-score that separates the bottom 85% from the top 15%, or the 85th percentile.

To find this, we need to find \( z \) such that \( P(Z < z) = 0.85 \).

From the z-table or using software:
- \( z \approx 1.04 \) when \( P(Z < z) = 0.85 \).

So, the answer is:
- **\( z \approx 1.04 \)** (rounded to 2 decimal places).

---

Would you like more detailed explanations on any part of this solution?

---

### Follow-Up Questions

1. What is the process to find z-scores for any given percentile?
2. How does one interpret a z-score in a standard normal distribution?
3. Why is the z-score for \( P(Z > z) = 0.8 \) negative in part (ii)?
4. How are cumulative probabilities used in z-score calculations?
5. How would these calculations change if the distribution were not standard normal?

### Tip
When using a z-table, remember that it typically gives cumulative probabilities from the left up to a z-score. Adjust accordingly for right-tail probabilities by subtracting from 1.

Find the indicated z-scores based on given probabilities.

Learn how to find z-scores based on specific probabilities, including calculations for cumulative probabilities and percentile ranks, such as the 85th percentile. Perfect for college statistics students or advanced high school learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation