sketch and find the area To the right of z = 0.56 and to the left of z = 0

To solve the problem of finding the area under the standard normal curve to the right of \( z = 0.56 \) and to the left of \( z = 0 \), we'll follow these steps:

### Step 1: Sketch the Area
1. **Draw the standard normal distribution curve**, which is a bell-shaped curve centered at \( z = 0 \).
2. **Mark the points** \( z = 0 \) and \( z = 0.56 \) on the horizontal axis.
3. **Shade the area** between \( z = 0 \) and \( z = 0.56 \). This is the area we need to find.

### Step 2: Find the Area Using Z-Table
The area under the standard normal curve between two z-values can be found using the standard normal distribution table (Z-table).

1. **Area to the left of \( z = 0 \)**: The Z-table gives the cumulative area to the left of a z-value. For \( z = 0 \), the cumulative area is 0.5.
2. **Area to the left of \( z = 0.56 \)**: Look up \( z = 0.56 \) in the Z-table, which corresponds to a cumulative area of approximately 0.7123.

### Step 3: Calculate the Area Between the Two Z-values
To find the area between \( z = 0 \) and \( z = 0.56 \), subtract the cumulative area at \( z = 0 \) from the cumulative area at \( z = 0.56 \):

\[
\text{Area} = \text{Area to the left of } z = 0.56 - \text{Area to the left of } z = 0 = 0.7123 - 0.5 = 0.2123
\]

### Final Answer:
The area to the right of \( z = 0 \) and to the left of \( z = 0.56 \) is **0.2123**.

Would you like a detailed explanation or have any further questions?

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Here are five related questions for further exploration:
1. How do you find the area under the curve for a negative z-value?
2. What is the total area under the standard normal curve?
3. How would you find the area to the right of a specific z-value?
4. What is the significance of a z-score in statistics?
5. How would you calculate the area between two negative z-scores?

**Tip:** Remember that the total area under the standard normal distribution curve is always 1 (or 100%), which represents the entire probability space.

This guide explains how to find the area under the standard normal curve between z = 0 and z = 0.56. It includes a step-by-step solution using Z-tables, suitable for students in Grades 10-12.

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