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Suppose Z follows the standard normal distribution. Use the calculator provided, or this table, to determine the value of c so that the following is true. Carry your intermediate computations to at least four decimal places. Round your answer to two decimal places.

To solve this problem, we need to find the value of \(c\) such that the probability \( P(-0.55 \leq Z \leq c) = 0.6820 \) for a standard normal distribution \(Z\).

### Step 1: Understand the problem
The probability \( P(-0.55 \leq Z \leq c) = 0.6820 \) tells us that the area under the standard normal curve between \( Z = -0.55 \) and \( Z = c \) is 0.6820.

### Step 2: Find the cumulative probability up to \( Z = -0.55 \)
First, let's find the cumulative probability up to \( Z = -0.55 \) using the standard normal distribution table or a calculator:

\[
P(Z \leq -0.55) \approx 0.2912
\]

### Step 3: Calculate the cumulative probability up to \( Z = c \)
We know that the total probability for \( -0.55 \leq Z \leq c \) is 0.6820. Therefore, the cumulative probability up to \( Z = c \) is:

\[
P(Z \leq c) = P(Z \leq -0.55) + 0.6820
\]

Substituting the value we found:

\[
P(Z \leq c) = 0.2912 + 0.6820 = 0.9732
\]

### Step 4: Find the z-score corresponding to the cumulative probability
Now, we need to find the z-score \(c\) such that \( P(Z \leq c) = 0.9732 \).

Using the standard normal distribution table or a calculator, we find:

\[
c \approx 1.94
\]

### Conclusion:
The value of \(c\) that satisfies the given condition is approximately **1.94**.

Would you like more details, or do you have any questions?

Here are 8 related questions that might interest you:
1. What is the meaning of the cumulative probability in a normal distribution?
2. How do you find the z-score for a given cumulative probability?
3. What is the probability for \( Z \) between 0 and 1.94?
4. How would you calculate the probability for \( Z \) between two arbitrary points?
5. How does the standard normal distribution differ from other normal distributions?
6. Can you calculate the value of \( c \) if the probability was \( 0.75 \) instead of \( 0.6820 \)?
7. What is the significance of the value \( Z = 0 \) in the standard normal distribution?
8. How would you explain the use of the normal distribution in real-world scenarios?

**Tip:** When working with standard normal distributions, it's helpful to understand how to read and interpret z-tables or use a calculator to find cumulative probabilities efficiently.

Learn how to find the value of c in a standard normal distribution using cumulative probabilities. Step-by-step solution provided with detailed calculations and rounding rules. Suitable for advanced high school and college students.

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