Find the area under the normal curve in each of the following
cases with draw 1. to the right of z = 1.27 
2. to the left of z = -1.32
3. between z = -2.35 and z = -1.47
4. between z = -0.72 and z = 1.35
5. to the right of z = - 0.90

To solve these problems, we need to calculate the area under the standard normal curve for various values of the z-score. These areas correspond to probabilities, which can be found using the standard normal distribution table or a calculator. Let's address each case:

1. **Area to the right of \( z = 1.27 \):**
   - The area to the right of a z-score is calculated as \( 1 - P(Z < 1.27) \).
   - From the z-table, \( P(Z < 1.27) \approx 0.8980 \).
   - So, the area is:  
     \[
     1 - 0.8980 = 0.1020
     \]
   The area to the right of \( z = 1.27 \) is **0.1020**.

2. **Area to the left of \( z = -1.32 \):**
   - From the z-table, \( P(Z < -1.32) \approx 0.0934 \).
   The area to the left of \( z = -1.32 \) is **0.0934**.

3. **Area between \( z = -2.35 \) and \( z = -1.47 \):**
   - We need to calculate \( P(-2.35 < Z < -1.47) \), which is \( P(Z < -1.47) - P(Z < -2.35) \).
   - From the z-table, \( P(Z < -2.35) \approx 0.0094 \) and \( P(Z < -1.47) \approx 0.0708 \).
   - So, the area is:
     \[
     0.0708 - 0.0094 = 0.0614
     \]
   The area between \( z = -2.35 \) and \( z = -1.47 \) is **0.0614**.

4. **Area between \( z = -0.72 \) and \( z = 1.35 \):**
   - We calculate \( P(-0.72 < Z < 1.35) \), which is \( P(Z < 1.35) - P(Z < -0.72) \).
   - From the z-table, \( P(Z < 1.35) \approx 0.9115 \) and \( P(Z < -0.72) \approx 0.2358 \).
   - So, the area is:
     \[
     0.9115 - 0.2358 = 0.6757
     \]
   The area between \( z = -0.72 \) and \( z = 1.35 \) is **0.6757**.

5. **Area to the right of \( z = -0.90 \):**
   - The area to the right of a z-score is calculated as \( 1 - P(Z < -0.90) \).
   - From the z-table, \( P(Z < -0.90) \approx 0.1841 \).
   - So, the area is:
     \[
     1 - 0.1841 = 0.8159
     \]
   The area to the right of \( z = -0.90 \) is **0.8159**.

Let me know if you would like to see any more detailed steps or have questions!

### Related Questions:
1. How do you find the area between any two z-scores using a z-table?
2. What is the total area under the normal curve, and why is it significant?
3. Can you explain the concept of z-scores in more depth?
4. How do negative z-scores differ from positive z-scores in the standard normal distribution?
5. What are some real-world applications of finding areas under the normal curve?

### Tip:
Always remember that the total area under the standard normal curve is 1, meaning it represents 100% of the probability distribution.

Find the area under the normal curve in each of the following cases with draw 1. to the right of z = 1.27 2. to the left of z = -1.32 3. between z = -2.35 and z = -1.47 4. between z = -0.72 and z = 1.35 5. to the right of z = -0.90

Learn how to calculate the area under the standard normal curve using z-scores. This guide covers key statistics concepts, including finding areas to the right and left of z-scores, as well as between two z-values. Suitable for Grades 10-12.

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