suppose a normal distrubtion has a mean of 25 and a standard deviation of 2. what is the area under the curve between 23 and 27

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To find the area under the curve between the values of 23 and 27 in a normal distribution with a mean of 25 and a standard deviation of 2, we can use the z-score formula and the standard normal distribution table.

1. Calculate the z-score for the lower boundary value of 23:
z1 = (23 - 25) / 2 = -1

2. Calculate the z-score for the upper boundary value of 27:
z2 = (27 - 25) / 2 = 1

3. Look up the corresponding probabilities for the z-scores in the standard normal distribution table:
P(Z < -1) ≈ 0.1587 (from the table)
P(Z < 1) ≈ 0.8413 (from the table)

4. Find the area between the two z-scores:
Area = P(Z < 1) - P(Z < -1)
= 0.8413 - 0.1587
≈ 0.6826

Therefore, the area under the curve between 23 and 27 in the given normal distribution is approximately 0.6826.

To find the area under the curve of a normal distribution between two values, you can use the concept of the standard normal distribution.

The standard normal distribution has a mean of 0 and a standard deviation of 1. However, the given normal distribution with a mean of 25 and a standard deviation of 2 can be converted to the standard normal distribution by applying the formula:

Z = (X - μ) / σ

Where Z is the z-score, X is the value in the original distribution, μ (mu) is the mean, and σ (sigma) is the standard deviation.

First, we need to convert the values 23 and 27 to their corresponding z-scores:

For 23:
Z1 = (23 - 25) / 2
Z1 = -2 / 2
Z1 = -1

For 27:
Z2 = (27 - 25) / 2
Z2 = 2 / 2
Z2 = 1

Now, we can use a standard normal distribution table or a calculator to find the area under the curve between -1 and 1. This represents the area under the curve between 23 and 27 in the original distribution.

Using a standard normal distribution table, you can find the corresponding area under the curve for Z = -1 and Z = 1. Subtracting the smaller area from the larger area gives us the desired result.

Note: The area under the standard normal distribution curve between -1 and 1 is approximately 0.6827.

Therefore, the area under the curve between 23 and 27 in the original distribution is approximately 0.6827.