X is normally distributed with a mean of 100 and standard deviation of 5. What is the probability that an event is greater than 90?

To solve this problem, we need to use the concept of Z-score in statistics. The Z-score is a measure of how many standard deviations an element is from the mean.

Step 1: Calculate the Z-score for the value 90. The formula for the Z-score is:

Z = (X - μ) / σ

where: X is the value from the dataset (in this case, 90), μ is the mean of the dataset (in this case, 100), and σ is the standard deviation of the dataset (in this case, 5).

So, Z = (90 - 100) / 5 = -2.

Step 2: Look up this Z-score in the Z-table to find the probability. The Z-table shows the cumulative probability from the mean up to a given Z-score.

For Z = -2, the cumulative probability is 0.0228, or 2.28%. However, this gives us the probability that an event is less than 90.

Step 3: Since we want the probability that an event is greater than 90, we need to subtract the cumulative probability from 1 (because the total probability of all events is 1).

So, the probability that an event is greater than 90 is 1 - 0.0228 = 0.9772, or 97.72%.