The Graduate Record Exam has combined verbal and quantitative mean of 1000 and a standard deviation of 200. Score range from 200 to 1600 and are approximately normally distributed.
a. What percentage score above 800?
b. Above what score do 20% of the test-takers score?
http://davidmlane.com/hyperstat/z_table.html
To answer these questions, we need to use the concept of the standard normal distribution, which allows us to convert any normal distribution to a standard normal distribution with a mean of 0 and a standard deviation of 1.
Let's start with question a:
a. To find the percentage of scores above 800, we need to calculate the z-score for a score of 800.
The formula to calculate the z-score is:
z = (X - μ) / σ
Where:
X is the value we want to find the z-score for (800 in this case),
μ is the mean (1000 in this case), and
σ is the standard deviation (200 in this case).
Substituting the values, we get:
z = (800 - 1000) / 200
z = -0.5
Now, we need to find the percentage of the distribution that is above a z-score of -0.5. We can use a standard normal distribution table or a calculator to find this value.
Looking up the z-score of -0.5 in the table or using a calculator, we find that approximately 30.85% of the scores fall below a z-score of -0.5.
Since we want the percentage of scores above 800, we subtract this percentage from 100%.
Percentage above 800 = 100% - 30.85% = 69.15%
Therefore, approximately 69.15% of the test-takers score above 800.
Now, let's move on to question b:
b. To determine the score above which 20% of the test-takers score, we need to determine the z-score associated with the 20th percentile.
The 20th percentile corresponds to a z-score such that 20% of the scores fall below it.
Using the standard normal distribution table or a calculator, we can find the z-score that corresponds to the 20th percentile. The z-score associated with the 20th percentile is approximately -0.84.
Now, we can use the z-score formula again to find the score that corresponds to a z-score of -0.84.
X = μ + z * σ
X = 1000 + (-0.84) * 200
X = 1000 - 168
X = 832
Therefore, approximately 20% of the test-takers score above 832.