Determine the area under the standard normal curve between z = - 1.04 and z = 2.76 (Round to the nearest

ten-thousandth, four decimal places.)

To find the area under the standard normal curve between \( z = -1.04 \) and \( z = 2.76 \), we need to calculate the probability \( P(-1.04 \leq Z \leq 2.76) \) for a standard normal distribution \( Z \sim N(0, 1) \).

1. **Find \( P(Z \leq 2.76) \):**  
   Using the standard normal table or a calculator, we get:
   \[
   P(Z \leq 2.76) \approx 0.9970
   \]

2. **Find \( P(Z \leq -1.04) \):**  
   Similarly, using the standard normal table:
   \[
   P(Z \leq -1.04) \approx 0.1492
   \]

3. **Calculate \( P(-1.04 \leq Z \leq 2.76) \):**  
   Subtract \( P(Z \leq -1.04) \) from \( P(Z \leq 2.76) \):
   \[
   P(-1.04 \leq Z \leq 2.76) = 0.9970 - 0.1492 = 0.8478
   \]

So, the area under the standard normal curve between \( z = -1.04 \) and \( z = 2.76 \) is approximately **0.8478**.

Would you like more details on this solution?

Here are some questions for further exploration:

1. How would you find the area under the normal curve between any two arbitrary \( z \)-values?
2. Why is it useful to use standard normal tables or calculators for these problems?
3. How do we interpret the area under the standard normal curve in real-world applications?
4. How would the area change if we used different \( z \)-values?
5. What steps would you take to find the area between \( z = 0 \) and \( z = 2.76 \)?

**Tip:** For quick calculations, use online tools or statistical software to find areas under the normal curve, especially for non-standard values.

Determine the area under the standard normal curve between z = -1.04 and z = 2.76 (Round to the nearest ten-thousandth, four decimal places.)

Find the area under the standard normal curve between z = -1.04 and z = 2.76 with this detailed solution. This problem includes probability calculations using the standard normal distribution, suitable for high school statistics students in Grades 10-12.

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