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 $P(Z > z)$).