To solve the problem of finding $z_0$ such that the area between $z_0$ and $z = -0.5$ is 0.2364, we need to refer to the standard normal distribution table (Z-table) or use statistical software to determine the values of $z_0$.

Steps:

1. Understanding the problem:

   • We are given the area between two z-scores: $z_0$ and $z = -0.5$.
   • The area between these z-scores is $0.2364$.

2. Area under the normal curve:

   • The total area under the normal distribution curve is 1.
   • A Z-table or a calculator provides the cumulative area from the left tail (i.e., from $z = -\infty$) up to a certain z-score.

3. Using the Z-table or normal distribution calculator:

   • The area to the left of $z = -0.5$ is about 0.3085 (from the Z-table).
   • We want the area between $z_0$ and $z = -0.5$ to be 0.2364. Thus, we need to find $z_0$ such that the difference between the cumulative area at $z_0$ and $z = -0.5$ gives 0.2364.

4. Cumulative area for $z_0$:

   • To find $z_0$, subtract the area 0.2364 from the cumulative area at $z = -0.5$: $\text{Cumulative area at } z_0 = 0.3085 - 0.2364 = 0.0721$
   • Now, we look for the z-score that corresponds to a cumulative area of 0.0721 in the Z-table.

5. From the Z-table:

   • The z-score corresponding to a cumulative area of 0.0721 is approximately $z_0 = -1.45$.

So, the two values of $z_0$ are approximately $-1.45$ and 0.11.

Thus, the answers are: $z_0 = -1.45 \quad \text{and} \quad z_0 = 0.11$

Would you like further details on the calculation? Here are five related questions to explore:

1. What is the significance of cumulative area in a normal distribution?
2. How can you find areas for other z-scores using a Z-table?
3. How does symmetry of the normal distribution help in solving such problems?
4. How do you interpret negative and positive z-scores in the context of the normal curve?
5. What is the difference between a cumulative area and the probability between two z-scores?

Tip: Always make sure to subtract the areas carefully, ensuring you’re using the correct cumulative area for each z-score when solving similar problems.