Find the percent of area under a normal curve between the mean and the given number of standard deviations from the mean.​ (Note that positive indicates above the​ mean, while negative indicates below the​ mean.)

0.400.40

To find the percent of the area under a normal curve between the mean and 0.40 standard deviations above the mean, we can use the standard normal distribution table or a calculator.

The z-score here is \( z = 0.40 \).

First, let's find the cumulative probability for \( z = 0.40 \).

Using the standard normal distribution table or a calculator, the cumulative probability for \( z = 0.40 \) is approximately 0.6554.

Since we want the area between the mean (which corresponds to \( z = 0 \)) and \( z = 0.40 \):

1. The cumulative probability for \( z = 0 \) (the mean) is 0.5.
2. The cumulative probability for \( z = 0.40 \) is 0.6554.

Now, subtract the cumulative probability at \( z = 0 \) from the cumulative probability at \( z = 0.40 \):

\[
0.6554 - 0.5 = 0.1554
\]

Therefore, the percent of the area under the normal curve between the mean and 0.40 standard deviations above the mean is \( 0.1554 \times 100 = 15.54\% \).

Would you like any more details or have any other questions?

Here are 5 questions you might want to ask next:
1. How do you interpret z-scores in a normal distribution?
2. What is the probability of a z-score being above a certain value?
3. How do you find the area between two z-scores using the standard normal distribution table?
4. What is the significance of the 68-95-99.7 rule in normal distributions?
5. How can you use the normal distribution in hypothesis testing?

**Tip:** When using the standard normal distribution, always remember that the total area under the curve is 1 (or 100%). This can help you quickly check if your calculations are reasonable.

Learn how to calculate the percent of area under a normal curve between the mean and 0.40 standard deviations above the mean. Understand the concept of z-scores and cumulative probabilities. Suitable for undergraduate level students and those interested in statistics and probability.

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