(a) Find z0 such that the area between z0 and z=-0.5 is 0.2364 (two answers). Round your answer to at least two decimal places. z0= ---- and 0.11

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.

Find z0 such that the area between z0 and z=-0.5 is 0.2364 (two answers). Round your answer to at least two decimal places. z0= ---- and 0.11

Learn how to find z0 such that the area between z0 and z = -0.5 is 0.2364 in the context of the standard normal distribution. This problem covers key statistical concepts like Z-scores and cumulative probabilities, with step-by-step solutions. Suitable for college-level statistics students.

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