To solve for $z$ in the probability statement $P(-z < Z < z) = 0.2468$, we need to understand that this represents the probability that a standard normal variable $Z$ lies within the symmetric interval $(-z, z)$ about the mean 0 of the standard normal distribution.

Steps:

1. Symmetry of the Standard Normal Distribution: Since the standard normal distribution is symmetric, the probability $P(-z < Z < z)$ corresponds to the cumulative probability from $-z$ to $z$, and the tails of the distribution (outside this interval) must sum up to the remaining probability.

2. Relation to the Standard Normal Table: The total probability under the normal curve is 1. If $P(-z < Z < z) = 0.2468$, then the probability in the tails (outside the interval $(-z, z)$) is: $1 - 0.2468 = 0.7532$ Since the distribution is symmetric, the probability in each tail is: $\frac{0.7532}{2} = 0.3766$

3. Using the Standard Normal Table: Now we look for the value of $z$ such that the cumulative probability from $-\infty$ to $z$ is $0.3766$. This corresponds to the cumulative distribution function (CDF) for $Z$.

4. Finding $z$: Using a standard normal distribution table or calculator, we find the z-value that gives a cumulative probability of 0.3766.

The z-value that corresponds to a cumulative probability of 0.3766 is approximately $z = -0.314$.

Thus, $z \approx 0.314$ (we consider the positive value since the interval is symmetric).

Final Answer:

The value of $z$ that satisfies $P(-z < Z < z) = 0.2468$ is approximately $z = 0.314$.

Would you like further details on this, or do you have any additional questions?

Here are some related questions:

1. What is the meaning of a standard normal distribution?
2. How do you use the z-table to find probabilities?
3. What is the relationship between probability and the cumulative distribution function (CDF)?
4. How do you calculate probabilities for non-standard normal distributions?
5. What is the significance of symmetric intervals in normal distribution problems?

Tip: When working with z-values and probabilities, always check whether the problem refers to a one-tailed or two-tailed scenario, as it affects the interpretation of the probability.