The scores on a psychology exam were normally distributed with a mean of 66 and a standard deviation of 5. A failing grade on the exam was anything 2 or more
standard deviations below the mean. What was the cutoff for a failing score? Approximately what percentage of the students failed?
To find the cutoff for a failing score, we need to find 2 standard deviations below the mean:
Cutoff = Mean - (2 * Standard Deviation)
Cutoff = 66 - (2 * 5)
Cutoff = 66 - 10
Cutoff = 56
Therefore, a failing score on the exam is anything below 56.
To find the percentage of students who failed, we need to find the area under the normal distribution curve to the left of the cutoff (56).
First, we need to calculate the z-score for the cutoff:
Z = (X - Mean) / Standard Deviation
Z = (56 - 66) / 5
Z = -10 / 5
Z = -2
After finding the z-score, we can use a standard normal distribution table or a calculator to find the percentage of students who scored below the cutoff.
From the standard normal distribution table, a z-score of -2 corresponds to an area of 0.0228.
Therefore, approximately 2.28% of the students failed the psychology exam.