Find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. About ____% of the area between z = -1 and z = 1 ( or within 1 standard deviation of the mean). DONT round answer

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.

Calculate the area under the standard normal distribution curve between z = -1 and z = 1. Learn how to find the cumulative probability and convert it into a percentage. This detailed solution demonstrates key statistical concepts and formulas. Suitable for high school students (Grades 10-12).

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