https://file.aiquickdraw.com/mathbot/file/1728801812865s54p2r2j.jpg

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.

Suppose Z follows the standard normal distribution. Find the value of c so that P(c ≤ Z ≤ -0.7) = 0.2019.

Learn how to find the value of c in the standard normal distribution where the probability P(c ≤ Z ≤ -0.7) = 0.2019. Explore the use of the cumulative distribution function (CDF) and z-scores in probability calculations. Suitable for students in introductory statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation