To find the probability for the standard normal distribution between $z = -1.99$ and $z = -0.23$, you can use a standard normal table (Z-table) or a calculator. The Z-table gives the cumulative probability from the left of the curve up to a given z-value.

1. Find the cumulative probability for $z = -1.99$: From the Z-table, $P(Z < -1.99) \approx 0.0233$.

2. Find the cumulative probability for $z = -0.23$: From the Z-table, $P(Z < -0.23) \approx 0.4090$.

3. Calculate the probability between the two values: $P(-1.99 < Z < -0.23) = P(Z < -0.23) - P(Z < -1.99)$ $P(-1.99 < Z < -0.23) = 0.4090 - 0.0233 = 0.3857$

So, $P(-1.99 < Z < -0.23) \approx 0.3857$.

Would you like more details or have any questions?

Relative questions:

1. What is the cumulative probability for $z = 1.99$?
2. How do you calculate the probability $P(Z > z)$ for a given z-value?
3. What is the probability between two positive z-values (e.g., $P(0.5 < Z < 1.5)$)?
4. How can you calculate probabilities using a calculator instead of a Z-table?
5. How do you interpret z-values in a real-world context (e.g., in height or test scores)?

Tip:

Always remember that the Z-table typically provides the area to the left of a z-value, so subtracting cumulative probabilities is key for finding the area between two z-values.