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The problem asks to find the value of \( a \) such that:

\[
P(Z > a) = 0.69
\]

where \( Z \sim N(0,1) \), meaning \( Z \) follows a standard normal distribution (with mean 0 and variance 1).

### Step-by-Step Solution:

1. **Relation to Standard Normal Distribution:**
   The probability \( P(Z > a) = 0.69 \) can be rewritten using the complementary rule of probabilities as:
   \[
   P(Z \leq a) = 1 - 0.69 = 0.31
   \]
   This means we are looking for the value of \( a \) such that the cumulative distribution function (CDF) of the standard normal distribution equals 0.31.

2. **Using the Z-table or Inverse CDF:**
   The value of \( a \) corresponds to the inverse of the standard normal CDF for a probability of 0.31. Using a Z-table or a calculator that computes the inverse of the CDF for a standard normal distribution, we find that:
   \[
   a \approx -0.4973
   \]

Thus, the value of \( a \) is approximately \( -0.4973 \).

---

Would you like more details on how this calculation is performed or have any further questions?

Here are 5 related questions to explore:

1. What is the relationship between the CDF and the PDF of the normal distribution?
2. How can you find probabilities for a normal distribution that is not standard (i.e., with different mean and variance)?
3. What is the difference between \( P(Z > a) \) and \( P(Z \leq a) \)?
4. How do you interpret probabilities in the context of a standard normal distribution?
5. What is the significance of a Z-score in statistics?

**Tip:** When working with normal distributions, always check if it's standard or not, as non-standard normal distributions require transformations before using Z-tables or inverse CDFs.

Let Z ~ N(0,1). Find a so that P(Z > a) = 0.69.

Learn how to find the value of a in a standard normal distribution where P(Z > a) = 0.69 with a detailed step-by-step solution. This problem covers key concepts in probability, normal distribution, and uses the complementary probability rule. Suitable for students in Grades 11-12.

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