A normal distribution has a mean of ƒÝ = 100 with ƒã = 20. If one score is randomly selected from this distribution, what is the probability that the score will have a value between X = 90 and X = 110?

a. 0.6915
b. 0.3085
c. 0.1915
d. 0.3830

z = (90-100)/20 = -.5

z table = 0.3085
z = (110-100)/20 = .5

z table = 0.6915

0.6915-0.3085 = 0.3830
D

To find the probability that a score will have a value between X = 90 and X = 110 in a normal distribution with a mean of ƒÝ = 100 and standard deviation of ƒã = 20, we need to use the standard normal distribution.

Step 1: Calculate the z-scores for X = 90 and X = 110.

To calculate the z-score, we use the formula:
z = (X - ƒÝ) / ƒã

For X = 90:
z = (90 - 100) / 20
z = -0.5

For X = 110:
z = (110 - 100) / 20
z = 0.5

Step 2: Use the z-score to find the probability.

We need to find the area under the standard normal distribution curve between z = -0.5 and z = 0.5.

Using a standard normal distribution table or a calculator, we can find that the probability corresponding to z = -0.5 is 0.3085, and the probability corresponding to z = 0.5 is 0.6915.

Step 3: Calculate the probability that the score will have a value between X = 90 and X = 110.

To find the probability between X = 90 and X = 110, we subtract the probability corresponding to z = -0.5 from the probability corresponding to z = 0.5.

Probability = 0.6915 - 0.3085
Probability = 0.3830

So, the correct answer is d. 0.3830.

To find the probability that a score will have a value between X = 90 and X = 110 in a normal distribution, we will use the properties of the standard normal distribution.

Step 1: Standardize the values
We need to standardize the values of X = 90 and X = 110 using the formula:

Z = (X - ƒÝ) / ƒã

Where Z is the standardized value, X is the value we want to find the probability for, ƒÝ is the mean, and ƒã is the standard deviation.
Substituting the given values, we get:

Z1 = (90 - 100) / 20 = -0.5
Z2 = (110 - 100) / 20 = 0.5

Step 2: Find the probability using the standard normal distribution table
We can now use the standard normal distribution table or a calculator to find the probability of Z being between -0.5 and 0.5.
From the table, we find that the probability for Z being between -0.5 and 0.5 is approximately 0.3830.

Therefore, the correct answer is d. 0.3830.