The image contains a problem that appears to be about determining the value of a variable given a probability and a normal distribution table.

Problem Statement:

You are likely asked to determine the value of a variable $Z$ (often referred to as the Z-score) associated with a certain probability in the context of the standard normal distribution.

General Approach:

1. Identify the Probability: The probability is given, and you need to find the corresponding Z-value from the standard normal distribution table.

2. Use the Z-table: The Z-table gives the area (or probability) to the left of a given Z-score in a standard normal distribution. Depending on the exact problem, you may need to find the Z-score corresponding to a cumulative probability.

3. Determine the Z-Score:

   • If the probability is given as $P(Z \leq z) = p$, find $z$ in the Z-table where the cumulative probability equals $p$.
   • If the probability is for the right tail $P(Z > z) = p$, you would find the Z-score corresponding to $1 - p$ in the table.

Example Calculation:

Let's say the problem gives you a probability of $P(Z \leq z) = 0.975$ and asks for the Z-value.

• Step 1: Look up the probability $0.975$ in the Z-table.
• Step 2: The closest value in the Z-table to $0.975$ is typically around $z = 1.96$.

Thus, the Z-score corresponding to $P(Z \leq z) = 0.975$ is approximately $z = 1.96$.

If you can specify the exact probability or further clarify the problem, I can provide a more tailored solution.

Do you have any questions or need more details on how to use the Z-table? Here are five related questions for further exploration:

1. How would you find the Z-score corresponding to a different cumulative probability, such as $P(Z \leq z) = 0.90$?
2. What is the Z-score if the cumulative probability is $P(Z \geq z) = 0.025$?
3. How would you calculate the probability for a non-standard normal distribution?
4. Can you explain the relationship between Z-scores and standard deviations in the normal distribution?
5. How does the shape of the normal distribution change as the mean or standard deviation changes?

Tip: When working with Z-tables, remember that they typically provide the cumulative probability from the far left of the distribution up to a specific Z-score.