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To solve these problems involving the standard normal distribution, we’ll use the standard normal (Z) table or a calculator to find the probabilities.

### Problem Breakdown
1. **(a) \( P(Z \leq -1.58) \)**:
   - This represents the probability that the Z-score is less than or equal to -1.58.
   - From the Z-table, find the cumulative probability associated with \( Z = -1.58 \).

2. **(b) \( P(Z > 0.74) \)**:
   - This represents the probability that the Z-score is greater than 0.74.
   - Find \( P(Z \leq 0.74) \) using the Z-table, then subtract from 1 to find the probability for \( P(Z > 0.74) \).

3. **(c) \( P(-0.97 < Z < 2.12) \)**:
   - This is the probability that Z is between -0.97 and 2.12.
   - Find \( P(Z \leq 2.12) \) and \( P(Z \leq -0.97) \) separately using the Z-table, then subtract the latter from the former to find the desired probability.

Let’s calculate each:

### Solutions
1. **(a) \( P(Z \leq -1.58) \)**:
   - From the Z-table, \( P(Z \leq -1.58) = 0.0571 \).

2. **(b) \( P(Z > 0.74) \)**:
   - From the Z-table, \( P(Z \leq 0.74) = 0.7704 \).
   - Therefore, \( P(Z > 0.74) = 1 - 0.7704 = 0.2296 \).

3. **(c) \( P(-0.97 < Z < 2.12) \)**:
   - From the Z-table, \( P(Z \leq 2.12) = 0.9830 \) and \( P(Z \leq -0.97) = 0.1660 \).
   - Therefore, \( P(-0.97 < Z < 2.12) = 0.9830 - 0.1660 = 0.8170 \).

### Final Answers
- (a) \( P(Z \leq -1.58) = 0.057 \)
- (b) \( P(Z > 0.74) = 0.230 \)
- (c) \( P(-0.97 < Z < 2.12) = 0.817 \)

Would you like further details on any part of these calculations?

Here are some related questions to expand your understanding:
1. What is a Z-score and how is it calculated?
2. How do you interpret cumulative probabilities in the standard normal distribution?
3. How would you find probabilities if given a non-standard normal distribution?
4. How can you calculate probabilities using software tools like Excel or online calculators?
5. What is the significance of rounding probabilities to three decimal places?

**Tip:** When calculating probabilities, remember that the total area under the standard normal distribution curve is always 1, which helps in finding complementary probabilities (like \( P(Z > z) \)).

Suppose Z follows the standard normal distribution. Calculate the following probabilities using the ALEKS calculator: (a) P(Z ≤ -1.58), (b) P(Z > 0.74), (c) P(-0.97 < Z < 2.12).

Learn how to calculate probabilities for Z-scores in a standard normal distribution. This includes P(Z ≤ -1.58), P(Z > 0.74), and P(-0.97 < Z < 2.12), using key concepts like cumulative probabilities and the Z-table. Suitable for college-level statistics students.

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