To solve each part of this problem, we need to use the standard normal distribution table (Z-table) or an inverse cumulative distribution function (often denoted as $Z^{-1}$) to find the corresponding $z_0$ values that satisfy each given probability.

Let's go through each part:

Part (a)

• Given: $P(Z \leq z_0) = 0.85$
• Find: $z_0$

For this, we need the Z-value that corresponds to a cumulative probability of 0.85. Checking a Z-table or using an inverse function, we find that: $z_0 \approx 1.036$

Part (b)

• Given: $P(Z \leq z_0) = 0.18$
• Find: $z_0$

We look up 0.18 in the Z-table or use the inverse function: $z_0 \approx -0.915$

Part (c)

• Given: $P(Z \geq z_0) = 0.3$
• Find: $z_0$

Since $P(Z \geq z_0) = 0.3$, we have $P(Z \leq z_0) = 1 - 0.3 = 0.7$. We then find the Z-value corresponding to 0.7: $z_0 \approx 0.524$

Part (d)

• Given: $P(0 \leq Z \leq 0.12) = 0.481$
• Find: $z_0$

Here, we are given the probability between 0 and $z_0 = 0.12$ and we need to match it with 0.481. We will need to confirm this setup because the usual Z-scores for cumulative values do not typically produce 0.481 between standard values. If the interpretation of this question requires further clarification, it may need numerical integration or specific computation. Let me know if you'd like me to double-check this step or provide further guidance on handling complex integrals in normal distribution contexts.

Would you like detailed explanations for each step, or do you have further questions?

Related Questions

1. How is the Z-score calculated for probabilities that are not exact values in the Z-table?
2. What is the interpretation of a negative Z-score, and when does it occur?
3. How does the cumulative distribution function (CDF) of the normal distribution differ from the probability density function (PDF)?
4. How do we handle probabilities for non-symmetric intervals in the Z-table?
5. What is the significance of the Z-score in standard normal distribution and its practical applications?

Tip

Remember that when calculating probabilities involving normal distributions, you can also use statistical software or calculators that provide inverse cumulative distribution functions for faster results.