The problem asks us to find the value of $c$ for which the probability:

$P(c \leq Z \leq -0.7) = 0.2019$

is true, where $Z$ follows the standard normal distribution.

Step-by-step process:

1. Standard Normal Distribution Concept: The standard normal distribution, $Z$, has a mean of 0 and a standard deviation of 1. We are interested in finding $c$, which corresponds to a probability of 0.2019 between $c$ and -0.7.

2. Using the Cumulative Distribution Function (CDF): The probability between two values under the standard normal distribution can be found by subtracting the CDF values: $P(c \leq Z \leq -0.7) = \Phi(-0.7) - \Phi(c)$ where $\Phi(x)$ is the CDF of the standard normal distribution.

3. Find ( \Phi(-0.7) : Using standard normal distribution tables or a calculator, we find: $\Phi(-0.7) \approx 0.2420$

4. Set up the equation: $0.2420 - \Phi(c) = 0.2019$ Solving for $\Phi(c)$: $\Phi(c) = 0.2420 - 0.2019 = 0.0401$

5. Find $c$ from the CDF: Using the inverse of the CDF (also known as the z-score table), $\Phi^{-1}(0.0401) \approx -1.75$.

Thus, the value of $c$ is approximately -1.75.

Final Answer:

The value of $c$ is $-1.75$ (rounded to two decimal places).

Would you like more details or further clarification on any step? Here are some related questions you can explore:

1. How do we interpret the CDF for a standard normal distribution?
2. What is the relationship between probability and the area under the normal curve?
3. How do z-scores relate to probabilities in a normal distribution?
4. Can this process be generalized for any probability between two points on a normal distribution?
5. How do we use the inverse of the normal distribution to find z-scores?

Tip: Always check the normal distribution table carefully, especially when using it for cumulative probabilities and inverse lookups.