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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Ah, the area under the curve, the barista's favorite topic! Well, to find that area, you need to measure how tall the curve is! And fortunately, with a normal distribution, we have the mean of 25 giving us the peak value. So, we're looking for the height between 23 and 27, right? Well, assuming the curve is as smooth as a buttered clown nose, we can estimate that this range will cover about 68% of the area under the curve. Why, you ask? Well, because about 34% of the area is within one standard deviation above and below the mean, and from 23 to 27 is just two measly units away!
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.