A standardized test is designed so that scores have a mean of 50 and a standard deviation of 4. What is the probability that a test score is above 54?
To find the probability that a test score is above 54, we first need to standardize the score using z-scores. The z-score measures how many standard deviations above or below the mean a particular score is.
The formula for calculating the z-score is:
z = (x - μ) / σ
Where:
z is the z-score
x is the value we want to standardize (in this case, 54)
μ is the mean (50)
σ is the standard deviation (4)
Using the given values, we can calculate the z-score:
z = (54 - 50) / 4
= 4 / 4
= 1
Now we need to look up the probability associated with this z-score in the standard normal distribution table (also known as the z-table). The standard normal distribution has a mean of 0 and a standard deviation of 1.
The z-table provides the cumulative probability up to a given z-score. Since we are looking for the probability that a score is above 54, we need to find the probability that a score is below 54 and subtract it from 1. In other words, we want 1 minus the cumulative probability up to the z-score of 1.
Looking up the z-score of 1 in the z-table, we find that the cumulative probability up to 1 is approximately 0.8413. Therefore, the probability that a test score is above 54 is:
1 - 0.8413 = 0.1587 (or approximately 15.9%)
So, the probability that a test score is above 54 is approximately 0.1587 or 15.9%.