The problem asks to find the area under the curve of the standard normal distribution between $z = -1$ and $z = 1$, which corresponds to the area within 1 standard deviation of the mean. Then, it asks to convert this area into a percentage.

Solution:

For a standard normal distribution (mean 0, standard deviation 1), the area between $z = -1$ and $z = 1$ is a well-known result in statistics.

1. From the empirical rule (or 68-95-99.7 rule), we know that approximately 68% of the data lies within 1 standard deviation of the mean in a normal distribution.

This is the percentage area under the curve between $z = -1$ and $z = 1$.

Thus, the answer for the blank is:

• About 68% of the area lies between $z = -1$ and $z = 1$.

Would you like more details on the empirical rule or any other related concepts?

Follow-up Questions:

1. What is the percentage of the area within 2 standard deviations of the mean?
2. How do we calculate the exact area between $z = -1$ and $z = 1$ using the standard normal table?
3. What is the area under the standard normal curve beyond $z = 2$?
4. How does the standard deviation affect the shape of a normal distribution?
5. Can you explain why the total area under the normal distribution curve is 1?

Tip:

Always remember that the empirical rule gives you quick approximations for 1, 2, and 3 standard deviations from the mean: 68%, 95%, and 99.7% respectively.