About % of the area under the curve of the standard normal distribution is between z=−2.398z=-2.398 and z=2.398z=2.398 (or within 2.398 standard deviations of the mean)

To find the percentage of the area under the curve of the standard normal distribution that lies between \( z = -2.398 \) and \( z = 2.398 \), we need to use the cumulative distribution function (CDF) of the standard normal distribution.

The standard normal distribution is symmetric about the mean (which is 0), and the CDF gives us the area to the left of a given \( z \)-score.

1. Calculate the cumulative probability for \( z = 2.398 \).
2. Calculate the cumulative probability for \( z = -2.398 \).
3. Subtract the cumulative probability of \( z = -2.398 \) from that of \( z = 2.398 \) to find the area between these two \( z \)-scores.

Using standard normal distribution tables or a statistical software, we get:

\[ P(Z \leq 2.398) \approx 0.9916 \]
\[ P(Z \leq -2.398) \approx 0.0084 \]

The area between \( z = -2.398 \) and \( z = 2.398 \) is:

\[ P(-2.398 \leq Z \leq 2.398) = P(Z \leq 2.398) - P(Z \leq -2.398) \]
\[ P(-2.398 \leq Z \leq 2.398) = 0.9916 - 0.0084 = 0.9832 \]

Therefore, about \( 98.32\% \) of the area under the curve of the standard normal distribution lies between \( z = -2.398 \) and \( z = 2.398 \).

Would you like further details on this calculation or have any other questions?

Here are 8 related questions for further exploration:
1. What is the standard normal distribution?
2. How is the cumulative distribution function (CDF) of the standard normal distribution used?
3. What is the significance of the \( z \)-score in statistics?
4. How do you find the area under the curve for different \( z \)-scores?
5. What is the empirical rule for the standard normal distribution?
6. How do you use statistical tables to find cumulative probabilities?
7. Can you explain the concept of symmetry in the standard normal distribution?
8. How does the central limit theorem relate to the standard normal distribution?

**Tip:** When using standard normal distribution tables, ensure that you understand whether the table gives the area to the left or right of the \( z \)-score, as this will affect your calculations.

Explore how to calculate the percentage of area under the curve of the standard normal distribution between z = -2.398 and z = 2.398 using cumulative distribution function (CDF). Detailed step-by-step solution and cumulative probabilities provided.

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