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.

---

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.