To find \( P(-0.2 \leq Z \leq 0.75) \), we need to calculate the probability that the standard normal variable \( Z \) falls between -0.2 and 0.75. The standard normal distribution, \( Z \), has a mean of 0 and a standard deviation of 1.

We can use the cumulative distribution function (CDF) of the standard normal distribution to find the probabilities for each bound separately:

1. \( P(Z \leq 0.75) \)
2. \( P(Z \leq -0.2) \)

The probability for the given range is then found by subtracting the smaller cumulative probability from the larger one:
\[
P(-0.2 \leq Z \leq 0.75) = P(Z \leq 0.75) - P(Z \leq -0.2)
\]

Using standard normal distribution tables or a calculator:

- \( P(Z \leq 0.75) \approx 0.7734 \)
- \( P(Z \leq -0.2) \approx 0.4207 \)

Thus, the probability is:
\[
P(-0.2 \leq Z \leq 0.75) = 0.7734 - 0.4207 = 0.3527
\]

So, \( P(-0.2 \leq Z \leq 0.75) \approx 0.3527 \).

Would you like more details or have any questions?

Here are 5 related questions to expand this topic:

1. How do you calculate probabilities for other ranges in a standard normal distribution?
2. What is the significance of the cumulative distribution function (CDF) in probability theory?
3. How can you interpret Z-scores and their relationship to probabilities?
4. What are the steps for finding probabilities outside a given range, like \( P(Z > 0.75) \)?
5. How do you use Z-tables to find probabilities manually?

**Tip:** To calculate probabilities between two Z-scores, always subtract the smaller cumulative probability from the larger one.

Learn how to calculate the probability P(-0.2 ≤ Z ≤ 0.75) using the cumulative distribution function (CDF) of the standard normal distribution. This detailed solution walks you through finding probabilities for Z-scores with step-by-step explanations and calculations. Suitable for students in Grades 11-12 and college-level statistics.

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