To solve $P(z < -2.31 \text{ or } z > -1.09)$ for a standard normal distribution, we can break it down into two parts:

1. $P(z < -2.31)$
2. $P(z > -1.09)$

Step 1: Find $P(z < -2.31)$

Using the standard normal distribution table or a calculator, we can find the cumulative probability for $z = -2.31$:

$P(z < -2.31) \approx 0.0104$

Step 2: Find $P(z > -1.09)$

For $P(z > -1.09)$, we first find the cumulative probability for $z = -1.09$:

$P(z < -1.09) \approx 0.1379$

Since $P(z > -1.09) = 1 - P(z < -1.09)$:

$P(z > -1.09) = 1 - 0.1379 = 0.8621$

Step 3: Combine the two parts

Now, we combine the two probabilities:

$P(z < -2.31 \text{ or } z > -1.09) = P(z < -2.31) + P(z > -1.09)$

$P(z < -2.31 \text{ or } z > -1.09) = 0.0104 + 0.8621 = 0.8725$

Final Answer:

$P(z < -2.31 \text{ or } z > -1.09) \approx 0.873$

Would you like more details, or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you interpret a standard normal distribution curve?
2. What is the significance of the z-score in probability?
3. How do you calculate the cumulative probability for any z-score?
4. What is the probability for the complement event $P(z > 2.31 \text{ and } z < -1.09)$?
5. How does the empirical rule (68-95-99.7) relate to this problem?

Tip: You can use z-tables, a calculator, or statistical software to find probabilities for any z-score in a standard normal distribution.