To find the area between $Z = 1.257$ and $Z = 2.501$ under the standard normal curve, we need to calculate the cumulative area for both $Z$-scores using a standard normal distribution table or technology (such as a calculator or software).

The cumulative area to the left of a $Z$-score gives the probability that a standard normal random variable is less than that value.

1. Find the cumulative area for $Z = 2.501$.
2. Find the cumulative area for $Z = 1.257$.
3. Subtract the cumulative area for $Z = 1.257$ from the area for $Z = 2.501$ to get the area between them.

Using technology (e.g., a Z-table calculator or statistical software), the results are:

• The cumulative area to the left of $Z = 2.501$ is approximately 0.9938.
• The cumulative area to the left of $Z = 1.257$ is approximately 0.8960.

Now, subtract these values to find the area between the two $Z$-scores:

$\text{Area} = 0.9938 - 0.8960 = 0.0978$

Thus, the area between $Z = 1.257$ and $Z = 2.501$ is approximately 0.0978.

Would you like more details on how to interpret these results or a breakdown of the process?

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Here are some related questions for further practice:

1. What is the area between $Z = 0.5$ and $Z = 1.5$ under the standard normal curve?
2. How would you find the area to the right of $Z = 1.96$?
3. What is the area between $Z = -1.5$ and $Z = 0.5$?
4. How do you calculate the area to the left of $Z = -1.25$?
5. What is the area between $Z = -2.33$ and $Z = -1.28$?

Tip: When using a standard normal table or technology, always ensure the values are rounded appropriately, especially when precision is crucial.