To find the area under the curve of the standard normal distribution between $z = -1$ and $z = 1$, we will use the properties of the cumulative distribution function (CDF) for the standard normal distribution.

The area between $z = -1$ and $z = 1$ is the difference between the cumulative probabilities at $z = 1$ and $z = -1$. Using the CDF values for a standard normal distribution:

$P(z \leq 1) = 0.8413447461$ $P(z \leq -1) = 0.1586552539$

Now, the area between $z = -1$ and $z = 1$ is:

$\text{Area} = P(z \leq 1) - P(z \leq -1) = 0.8413447461 - 0.1586552539 = 0.6826894922$

To convert this to a percentage, we multiply by 100:

$\text{Percentage} = 0.6826894922 \times 100 = 68.26894922\%$

Thus, about 68.26894922% of the area is between $z = -1$ and $z = 1$ (or within 1 standard deviation of the mean).

Would you like more details on this, or have any questions?

Here are 5 related questions:

1. What is the area under the curve between $z = -2$ and $z = 2$?
2. How do you calculate the area for non-standard normal distributions?
3. What percentage of the area lies outside of $z = -1$ and $z = 1$?
4. What does it mean for a normal distribution to be “standard”?
5. How do you calculate the CDF for any $z$-value?

Tip: The standard normal distribution has a mean of 0 and a standard deviation of 1, making it a great reference for probabilities in other normal distributions.