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Let’s solve these problems step by step. Since \( Z \) is a standard normal variable, it has a mean of 0 and a standard deviation of 1. We will use the standard normal distribution table (Z-table) to find the probabilities.

### 1. \( P(0 \leq Z < 1.97) \)
To find this probability, we first find the cumulative probability up to \( Z = 1.97 \) and subtract the cumulative probability up to \( Z = 0 \).

- From the Z-table, \( P(Z < 1.97) \approx 0.9756 \).
- \( P(Z < 0) = 0.5 \) since the standard normal distribution is symmetric about zero.

Thus, 
\[
P(0 \leq Z < 1.97) = P(Z < 1.97) - P(Z < 0) = 0.9756 - 0.5 = 0.4756.
\]

### 2. \( P(Z \geq 2.83) \)
To find this probability, we calculate \( 1 - P(Z < 2.83) \).

- From the Z-table, \( P(Z < 2.83) \approx 0.9977 \).

Therefore,
\[
P(Z \geq 2.83) = 1 - 0.9977 = 0.0023.
\]

### 3. \( P(Z < -1.77) \)
Since the Z-table gives values for positive Z, we can use the symmetry of the standard normal distribution: \( P(Z < -1.77) = P(Z > 1.77) \).

- From the Z-table, \( P(Z < 1.77) \approx 0.9616 \).

Thus,
\[
P(Z < -1.77) = 1 - 0.9616 = 0.0384.
\]

### 4. \( P(-1.23 < Z \leq 0) \)
We find the cumulative probability up to \( Z = 0 \) and subtract the cumulative probability up to \( Z = -1.23 \).

- \( P(Z < 0) = 0.5 \).
- From the Z-table, \( P(Z < -1.23) = 1 - P(Z < 1.23) \approx 1 - 0.8907 = 0.1093 \).

Therefore,
\[
P(-1.23 < Z \leq 0) = 0.5 - 0.1093 = 0.3907.
\]

Would you like further details on these calculations or have any questions?

### Related Questions:
1. How do you calculate probabilities for non-standard normal distributions?
2. What is the significance of the area under the standard normal curve?
3. How does the symmetry of the standard normal distribution simplify calculations?
4. How can you calculate probabilities for ranges not centered around zero?
5. What are the properties of a cumulative distribution function (CDF)?

### Tip:
When using Z-tables, always ensure that you understand whether the table provides cumulative probabilities from the left or if it gives tail probabilities.

If Z is a standard normal variable, find the following probabilities: P(0 ≤ Z < 1.97), P(Z ≥ 2.83), P(Z < -1.77), and P(-1.23 < Z ≤ 0).

Learn how to calculate probabilities using the standard normal distribution with specific Z-scores. This step-by-step guide explains how to find P(0 ≤ Z < 1.97), P(Z ≥ 2.83), P(Z < -1.77), and P(-1.23 < Z ≤ 0), using Z-tables and the properties of the normal distribution. Suitable for college students or advanced high school statistics.

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