https://file.aiquickdraw.com/mathbot/file/17290597213716s532hd9.jpg

The image shows a problem involving **normal distribution** with two tasks:

1. **First part**:  
   - You are asked to sketch the region corresponding to the statement \( P(-0.6 < z < -0.2) \).  
   - This involves shading the area between the two \( z \)-values, \( -0.6 \) and \( -0.2 \), on the standard normal curve.

2. **Second part**:  
   - You need to adjust the shaded region for the probability statement \( P(z < c) = 0.45 \).  
   - This asks for a left-tailed area under the normal curve where the cumulative probability is 0.45. Your task is to find the corresponding \( z \)-value (which will be negative) and shade the area to the left of it.

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### **Steps to solve:**

#### Part 1: \( P(-0.6 < z < -0.2) \)
1. You need to mark \( z = -0.6 \) and \( z = -0.2 \) on the standard normal curve.
2. The area between these two values (which are both negative) should be shaded.

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#### Part 2: \( P(z < c) = 0.45 \)
1. For this part, you need the **cumulative probability** value of 0.45, which corresponds to a specific \( z \)-score.
2. Using a **Z-table**, we find that \( z \approx -0.13 \) gives a cumulative probability of approximately 0.45.
3. You will shade the area to the **left of \( z = -0.13 \)**.

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If you apply these steps, the regions will correspond exactly to the probability ranges given in the task. Do you need more help with any part or further clarifications?

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### **Follow-up Questions:**
1. What does the cumulative probability of 0.45 represent in context?
2. How do we interpret negative \( z \)-scores in the normal distribution?
3. How would the shading change if the cumulative probability were 0.55 instead?
4. What is the total area under the standard normal curve?
5. How can you calculate the area between two \( z \)-values without a Z-table?

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**Tip**: When working with cumulative probabilities, always remember that the area under the curve to the left of a \( z \)-score gives the cumulative value.

Sketch the region corresponding to the statement P(-0.6 < z < -0.2) and P(z < c) = 0.45.

Understand how to sketch regions for probability calculations on the standard normal curve. Learn how to find areas for P(-0.6 < z < -0.2) and P(z < c) = 0.45, using the Z-table to determine the corresponding Z-scores. Suitable for high school students in Grades 10-12 studying statistics and probability.

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