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Let’s analyze the given problem!

You are asked to find the value of \( c \) such that:

\[
P(-1.04 \leq Z \leq c) = 0.8386
\]

This problem involves the **standard normal distribution**. Here's how we can solve it step-by-step:

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### Step 1: Break down the probabilities
- We know that \( Z \) follows the **standard normal distribution** \( N(0, 1) \), meaning it has a mean of 0 and standard deviation of 1.
- The probability \( P(-1.04 \leq Z \leq c) = 0.8386 \) represents the area under the standard normal curve between \( Z = -1.04 \) and \( Z = c \).

---

### Step 2: Use known areas from Z-tables
To solve for \( c \), we need the areas under the curve from the **Z-tables**.

1. **Find the cumulative probability at \( Z = -1.04 \)**:
   - From Z-tables, \( P(Z \leq -1.04) = 0.1492 \).

2. **Total area we want**:
   - The probability from \( Z = -1.04 \) to \( Z = c \) is given as 0.8386.

3. **Now solve for cumulative probability at \( Z = c \)**:
   - Let \( P(Z \leq c) \) be the cumulative probability we need.
   - From cumulative probabilities:
     \[
     P(Z \leq c) = P(Z \leq -1.04) + 0.8386
     \]
     \[
     P(Z \leq c) = 0.1492 + 0.8386 = 0.9878
     \]

---

### Step 3: Find the Z-value corresponding to 0.9878
- From Z-tables, the closest cumulative probability to 0.9878 corresponds to:
  \[
  Z \approx 2.25
  \]

---

### Step 4: Solution
Thus, the value of \( c \) that satisfies the equation is:

\[
c \approx 2.25
\]

---

Do you have any questions or need further clarification?

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Here are 5 related questions to deepen your understanding:

1. How do we interpret cumulative probabilities in a standard normal distribution?
2. What is the importance of Z-tables in solving probability problems?
3. How would the solution change if \( Z \) had a different mean or standard deviation?
4. What if the given probability was instead for \( P(Z \geq -1.04) \)?
5. How do we calculate probabilities for non-standard normal distributions?

**Tip**: When using Z-tables, always ensure you are looking up values with the correct cumulative probabilities starting from the left tail.

Suppose Z follows the standard normal distribution. Use the calculator or table to determine the value of c so that P(-1.04 ≤ Z ≤ c) = 0.8386.

Learn how to determine the Z-score 'c' in a standard normal distribution, given P(-1.04 ≤ Z ≤ c) = 0.8386. This problem involves calculating cumulative probabilities and interpreting Z-tables.

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