To solve for the value of $c$ such that $P(Z \leq c) = 0.8159$, where $Z$ follows the standard normal distribution, we'll follow these steps:

1. Understand the Problem:
   We need to find the value of $c$ where the cumulative probability up to $c$ is 0.8159. This requires using the standard normal distribution table (or calculator) to find the corresponding $c$-value.

2. Use the Standard Normal Table or Calculator:

   • The standard normal distribution table gives the probability $P(Z \leq c)$ for various values of $c$.
   • To find $c$, we locate 0.8159 in the table and then read off the corresponding $c$ value.

   However, since standard normal tables are usually organized to show values for up to 4 decimal places of probability, and this is slightly more direct using a calculator, I’ll directly compute this.

Let's calculate the value of $c$.

I'll now compute this.

The value of $c$ such that $P(Z \leq c) = 0.8159$ is $c = 0.90$ (rounded to two decimal places).

Would you like more details or have any questions about this?

Here are 5 related questions to explore:

1. What is the probability $P(Z \leq 1.5)$ for the standard normal distribution?
2. How would you find $P(Z \geq c) = 0.8159$?
3. What does it mean if $P(Z \leq c) = 0.5$?
4. How would the value of $c$ change if $P(Z \leq c) = 0.95$?
5. How can you use the standard normal distribution to find probabilities between two values?

Tip: Understanding how to read and use the standard normal table is crucial for finding probabilities and critical values in statistical analysis.