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The problem is asking to find the probability that a standard normal variable \( Z \) lies between \( -2.88 \) and \( 0.86 \). For a standard normal distribution, the mean is \( 0 \) and the standard deviation is \( 1 \). To solve this, we need to use the cumulative distribution function (CDF) values of the standard normal distribution.

The probability \( P(-2.88 \leq Z \leq 0.86) \) is calculated as:

\[
P(-2.88 \leq Z \leq 0.86) = P(Z \leq 0.86) - P(Z \leq -2.88)
\]

Using standard normal tables or a calculator:

- \( P(Z \leq 0.86) = 0.8051 \)
- \( P(Z \leq -2.88) = 0.0020 \)

Now subtract:

\[
P(-2.88 \leq Z \leq 0.86) = 0.8051 - 0.0020 = 0.8031
\]

Thus, the probability that \( Z \) lies between \( -2.88 \) and \( 0.86 \) is \( \boxed{0.8031} \).

Would you like further details or explanations on this?

Here are 5 related questions:
1. What is the general form for calculating probabilities with the standard normal distribution?
2. How do you interpret standard normal distribution probabilities in real-world contexts?
3. What is the significance of the cumulative distribution function (CDF) in probability?
4. How would you find the probability for other ranges in a normal distribution?
5. Can this concept be extended to non-standard normal distributions?

**Tip:** When working with standard normal distributions, always refer to the Z-tables or statistical calculators to find probabilities efficiently.

If Z is a standard normal variable, find the probability. The probability that Z lies between -2.88 and 0.86.

Learn how to calculate the probability that a standard normal variable Z lies between -2.88 and 0.86. This solution covers key concepts such as the cumulative distribution function (CDF) and properties of the standard normal distribution. Suitable for Grades 10-12.

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