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The problem in the image asks you to compute the probability for a standard normal variable \( Z \sim N(0,1) \) that \( 0.2 < Z < 0.9 \). This requires finding the area under the standard normal curve between these two values of \( Z \).

### Steps to solve:

1. **Use the CDF of the Standard Normal Distribution:** 
   The standard normal distribution table (or Z-table) provides the cumulative probabilities from the mean (0) up to a specific Z-value. For values like 0.2 and 0.9, you can use the table or a calculator that provides the cumulative distribution function (CDF) for a standard normal variable.

2. **CDF values:**
   - \( P(Z < 0.9) \) = CDF(0.9)
   - \( P(Z < 0.2) \) = CDF(0.2)

   The probability that \( Z \) lies between 0.2 and 0.9 is found by subtracting the CDF of 0.2 from the CDF of 0.9.

3. **Calculate the result:**
   Using standard normal CDF values:
   - \( P(Z < 0.9) = 0.8159 \)
   - \( P(Z < 0.2) = 0.5793 \)

   Therefore:
   \[
   P(0.2 < Z < 0.9) = P(Z < 0.9) - P(Z < 0.2) = 0.8159 - 0.5793 = 0.2366
   \]

Thus, the probability that \( Z \) lies between 0.2 and 0.9 is approximately **0.2366**.

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Do you want further details or have any questions?

Here are 5 related questions for deeper understanding:

1. How is the cumulative distribution function (CDF) of the standard normal distribution used in probability calculations?
2. What is the interpretation of the area under the curve in a normal distribution?
3. How can you compute probabilities for a non-standard normal distribution (i.e., not \( N(0, 1) \))?
4. What are Z-scores, and how are they related to the standard normal distribution?
5. How can the empirical rule (68-95-99.7 rule) be used to approximate probabilities for a normal distribution?

**Tip:** For small probability intervals, you can use symmetry properties of the normal distribution to simplify calculations.

Learn how to calculate P(0.2 < Z < 0.9) for a standard normal distribution N(0, 1). This solution demonstrates the use of the cumulative distribution function (CDF) to find the probability between two values in the Z-distribution.

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