To find the value of $c$ such that $P(0.52 \leq Z \leq c) = 0.2813$, where $Z$ follows the standard normal distribution, we can follow these steps:

1. Understand the Problem:

   • We know the probability between 0.52 and $c$ is 0.2813.
   • First, we'll find the cumulative probability $P(Z \leq 0.52)$ using the standard normal distribution.
   • Then, we'll add 0.2813 to this cumulative probability to find $P(Z \leq c)$.
   • Finally, we'll find the corresponding $c$-value for this cumulative probability.

2. Steps to Calculate $c$:

   • Calculate $P(Z \leq 0.52)$.
   • Add 0.2813 to this probability to find $P(Z \leq c)$.
   • Find the $c$-value corresponding to this cumulative probability.

Let's calculate this step by step.The value of $c$ such that $P(0.52 \leq Z \leq c) = 0.2813$ is $c = 2.05$ (rounded to two decimal places).

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How would you find $c$ if $P(-0.52 \leq Z \leq c) = 0.2813$?
2. What is $P(Z \leq 2.05)$ for the standard normal distribution?
3. How would the value of $c$ change if $P(0.52 \leq Z \leq c)$ were 0.35 instead of 0.2813?
4. What is the interpretation of a z-value of 2.05 in a standard normal distribution?
5. How would you solve for $c$ if the probability was given as $P(0 \leq Z \leq c) = 0.2813$?

Tip: Understanding how to manipulate probabilities and z-values is essential when working with the standard normal distribution, especially for hypothesis testing and confidence intervals.