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The problem asks to determine the value of \( c \) such that the probability \( P(c \leq Z \leq -0.69) = 0.2157 \), where \( Z \) follows a standard normal distribution.

### Step-by-Step Solution:

1. **Understand the probability range:**
   We need to find \( c \) such that the cumulative probability between \( c \) and \( -0.69 \) is 0.2157.

2. **Use the cumulative distribution function (CDF) of the standard normal distribution:**
   The probability \( P(c \leq Z \leq -0.69) = 0.2157 \) can be written as:
   \[
   P(Z \leq -0.69) - P(Z \leq c) = 0.2157
   \]
   We can break this down into two parts:
   - The cumulative probability \( P(Z \leq -0.69) \) can be found using a standard normal table or calculator.
   
3. **Find \( P(Z \leq -0.69) \):**
   Using the standard normal distribution table, we find:
   \[
   P(Z \leq -0.69) \approx 0.2451
   \]
   
4. **Set up the equation:**
   Now, using the known probability:
   \[
   0.2451 - P(Z \leq c) = 0.2157
   \]
   Solving for \( P(Z \leq c) \):
   \[
   P(Z \leq c) = 0.2451 - 0.2157 = 0.0294
   \]
   
5. **Determine \( c \):**
   From the standard normal distribution table, we find the z-score corresponding to a cumulative probability of 0.0294, which is approximately \( c = -1.89 \).

### Answer:
The value of \( c \) is approximately **-1.89**.

Would you like any further details or clarifications?

### 5 Related Questions:
1. How do we calculate the cumulative probability for any z-value using the standard normal distribution?
2. What is the significance of the standard normal distribution in statistics?
3. Can you explain the process of finding probabilities for ranges of z-values?
4. How does the value of \( c \) affect the cumulative probability?
5. How do we use z-scores to determine the probability for any given range?

### Tip:
Always ensure you're using at least four decimal places when dealing with probabilities in the standard normal distribution for precision, especially in exams or technical problems.

Suppose Z follows the standard normal distribution. Use the calculator or a table to determine the value of c so that the following is true: P(c ≤ Z ≤ -0.69) = 0.2157.

Learn how to determine the value of c such that the probability P(c ≤ Z ≤ -0.69) equals 0.2157 using standard normal distribution tables or a calculator. This problem demonstrates the use of Z-scores and cumulative probabilities in statistics.

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