To find the probability that a standard normal variable $Z$ lies between 0 and 3.01, we need to use the standard normal distribution, where the mean $\mu = 0$ and standard deviation $\sigma = 1$.

In the standard normal distribution table, or by using statistical software, we can find:

1. $P(Z \leq 3.01)$: This represents the cumulative probability up to 3.01.
2. $P(Z \leq 0)$: This represents the cumulative probability up to 0.

The probability that $Z$ lies between 0 and 3.01 is:

$P(0 < Z < 3.01) = P(Z \leq 3.01) - P(Z \leq 0)$

Looking up these values:

• $P(Z \leq 3.01) \approx 0.9987$
• $P(Z \leq 0) = 0.5$ (since 0 is the mean of the standard normal distribution)

Thus,

$P(0 < Z < 3.01) = 0.9987 - 0.5 = 0.4987$

Answer

The probability that $Z$ lies between 0 and 3.01 is approximately 0.4987.

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. What is a standard normal distribution, and why is it used in probability?
2. How can you interpret the probability values from a standard normal distribution table?
3. How would the calculation change if the question asked for $P(-3.01 < Z < 0)$?
4. Can we use software like Excel or Python to find cumulative probabilities for any Z value?
5. What does the cumulative distribution function (CDF) represent in a normal distribution?

Tip: When using standard normal tables, values are typically provided for positive Z-scores. Symmetry around the mean allows quick calculations for negative Z-scores.