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Let's go through each part of Questions 5 and 6, using the properties of the standard normal distribution (Z-distribution) and a Z-table to find the probabilities.

### Question 5
**Given:** Standard Normal Distribution, rounded to 3 decimal places.

1. **(a) \( P(Z > 1.96) \)**
   - Since \( Z \) is greater than a specific value, we need the area to the right of \( Z = 1.96 \).
   - The Z-table shows that \( P(Z < 1.96) = 0.975 \).
   - Thus, \( P(Z > 1.96) = 1 - 0.975 = 0.025 \).

2. **(b) \( P(Z < 0.8) \)**
   - We need the area to the left of \( Z = 0.8 \).
   - The Z-table gives \( P(Z < 0.8) = 0.7881 \).

3. **(c) \( P(-0.6 < Z < 1.2) \)**
   - We calculate \( P(Z < 1.2) \) and \( P(Z < -0.6) \) and subtract them.
   - From the Z-table:
     - \( P(Z < 1.2) = 0.8849 \)
     - \( P(Z < -0.6) = 0.2743 \)
   - Therefore, \( P(-0.6 < Z < 1.2) = 0.8849 - 0.2743 = 0.6106 \).

### Question 6
**Given:** Standard Normal Distribution, rounded to at least 4 decimal places.

1. **(a) \( P(Z > -0.23) \)**
   - We find \( P(Z < -0.23) \) and subtract from 1.
   - The Z-table shows \( P(Z < -0.23) = 0.4090 \).
   - So, \( P(Z > -0.23) = 1 - 0.4090 = 0.5910 \).

2. **(b) \( P(-2.7 < Z < 0.45) \)**
   - Find \( P(Z < 0.45) \) and \( P(Z < -2.7) \), then subtract.
   - From the Z-table:
     - \( P(Z < 0.45) = 0.6736 \)
     - \( P(Z < -2.7) = 0.0035 \)
   - Thus, \( P(-2.7 < Z < 0.45) = 0.6736 - 0.0035 = 0.6701 \).

3. **(c) \( P(Z < 1.57) \)**
   - From the Z-table, \( P(Z < 1.57) = 0.9418 \).

4. **(d) \( P(Z \geq 2.89) \)**
   - We find \( P(Z < 2.89) \) and subtract from 1.
   - From the Z-table, \( P(Z < 2.89) = 0.9981 \).
   - So, \( P(Z \geq 2.89) = 1 - 0.9981 = 0.0019 \).

### Summary of Answers

- **Question 5:**
  - (a) \( P(Z > 1.96) = 0.025 \)
  - (b) \( P(Z < 0.8) = 0.788 \)
  - (c) \( P(-0.6 < Z < 1.2) = 0.611 \)

- **Question 6:**
  - (a) \( P(Z > -0.23) = 0.5910 \)
  - (b) \( P(-2.7 < Z < 0.45) = 0.6701 \)
  - (c) \( P(Z < 1.57) = 0.9418 \)
  - (d) \( P(Z \geq 2.89) = 0.0019 \)

Would you like a deeper explanation of any step, or do you have any questions?

### Related Questions to Explore
1. How would the probabilities change if we used a different Z-score, like \( Z > 2.5 \)?
2. What are the steps to calculate probabilities without a Z-table?
3. How do you interpret a Z-score in real-world contexts?
4. What is the probability of a Z-score falling between two specific values (e.g., \( P(1 < Z < 2) \))?
5. Can we find probabilities for non-standard normal distributions?

### Tip
When using a Z-table, remember that it typically gives the area to the left of a given Z-score, so you may need to adjust your calculation for right-tail probabilities.

In a Standard Normal Distribution, find the probabilities for specific Z-values and compute various probabilities for a standard normal distribution, rounding to the required decimal places.

Learn how to calculate probabilities for Z-scores in a standard normal distribution. This guide covers right-tail, left-tail, and between-range probabilities using Z-tables. Examples include P(Z > 1.96), P(Z < 0.8), and more, with answers rounded to three and four decimal places.

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