In 4 decimal places, find the probability that a standard normal random variable Z falls between −2.00 and −0.44.

To find the probability that a standard normal random variable \( Z \) falls between \(-2.00\) and \(-0.44\), we need to calculate the cumulative distribution function (CDF) values for these two points and subtract them.

The standard normal distribution, denoted as \( Z \sim N(0, 1) \), has a mean of 0 and a standard deviation of 1.

### Step-by-Step Calculation:

1. **Find the CDF value for \( Z = -2.00 \):**

   The CDF value, \( \Phi(-2.00) \), represents the probability that the random variable \( Z \) is less than \(-2.00\).

2. **Find the CDF value for \( Z = -0.44 \):**

   Similarly, \( \Phi(-0.44) \) is the probability that \( Z \) is less than \(-0.44\).

3. **Calculate the probability \( P(-2.00 < Z < -0.44) \):**

   This probability can be found using the difference between the two CDF values:
   \[
   P(-2.00 < Z < -0.44) = \Phi(-0.44) - \Phi(-2.00)
   \]

Using the standard normal distribution table or a calculator to find the CDF values:

- \( \Phi(-2.00) \approx 0.0228 \)
- \( \Phi(-0.44) \approx 0.3300 \)

### Compute the Probability:

\[
P(-2.00 < Z < -0.44) = 0.3300 - 0.0228 = 0.3072
\]

Thus, the probability that a standard normal random variable \( Z \) falls between \(-2.00\) and \(-0.44\) is **0.3072**.

### Summary:

The probability \( P(-2.00 < Z < -0.44) \) to four decimal places is **0.3072**.

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Do you want more details on any part of this solution, or do you have any questions?

Here are 5 related questions for further exploration:

1. What is the probability that \( Z \) falls between 0 and 1 in a standard normal distribution?
2. How do you calculate the Z-score for a given data point?
3. What are the properties of the standard normal distribution?
4. How can we use the standard normal distribution to calculate probabilities for non-standard normal distributions?
5. What is the relationship between the normal distribution and the empirical rule?

**Tip:** Always remember that the area under the curve of a probability distribution function represents probabilities. For the standard normal distribution, this total area is equal to 1.

Learn how to calculate the probability that a standard normal random variable Z falls between -2.00 and -0.44 with a step-by-step solution. This problem covers the standard normal distribution, cumulative distribution function (CDF), and probability calculation. Suitable for advanced high school or college students.

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