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.