To find the probability $P(0 < Z < 2.582)$, where $Z$ represents a standard normal variable, we follow these steps:

1. Identify the cumulative distribution function (CDF) for $Z$: The CDF provides the probability that $Z$ is less than or equal to a given value.

2. Use the standard normal table or a calculator:

   • $P(Z < 2.582)$ can be obtained from the standard normal table or a calculator.
   • From the table or calculation, $P(Z < 2.582) \approx 0.9951$.

3. Subtract the probability for $P(Z < 0)$:

   • The standard normal distribution is symmetric about zero, so $P(Z < 0) = 0.5$.

4. Calculate the result: $P(0 < Z < 2.582) = P(Z < 2.582) - P(Z < 0)$ $P(0 < Z < 2.582) = 0.9951 - 0.5 = 0.4951$

Final Answer:

$P(0 < Z < 2.582) = 0.4951$

Would you like further clarification or a visual explanation?

Related Questions:

1. What is the probability of $P(Z < 2.582)$ without subtracting $P(Z < 0)$?
2. How do we find probabilities for non-standard normal distributions?
3. Can this process be used to find $P(Z > 2.582)$?
4. What is the probability of $P(-2.582 < Z < 2.582)$?
5. How does changing the standard deviation affect these probabilities?

Tip: Always double-check whether the standard normal distribution (mean = 0, standard deviation = 1) is being assumed or if a transformation is required.