4. The Graduate Record Exam (GRE) has a combined verbal and quantitative mean of 1000 and a standard deviation of 200. Scores range from 200 to 1600 and are approximately normally distributed. For each of the following problems:
(a) draw a rough sketch, darkening in the portion of the curve that relates to the answer, and
(b) indicate the percentage or score called for by the problem.
a. What percentage of the persons who take the test score above 1300?
b. What percentage score above 800?
c. What percentage score below 1200?
d. Above what score do 20% of the test-takers score?
e. Above what score do 30% of the test-takers score?
Z = (score-mean)/SD
a, b, c. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
d, e. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities mentioned. Insert Z score into equation above.
To answer these questions, we need to use the concept of standard deviation and z-scores. The formula to calculate a z-score is:
z = (x - μ) / σ
Where:
- x is the value we are interested in (the score)
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
First, let's calculate the z-scores for each of the questions:
a. What percentage of the persons who take the test score above 1300?
To find this, we need to determine the z-score for 1300. Given that the mean is 1000 and the standard deviation is 200:
z = (1300 - 1000) / 200 = 300 / 200 = 1.5
Now, we need to find the area under the right tail of the normal distribution curve corresponding to a z-score of 1.5. We can refer to a z-table to find this value. Looking up a z-table, we find that the area to the right of 1.5 is approximately 0.0668.
To determine the percentage, we can transform this value to a percentage:
Percentage = 0.0668 * 100% = 6.68%
So, approximately 6.68% of the persons who take the test score above 1300.
b. What percentage score above 800?
Similarly, let's find the z-score for 800:
z = (800 - 1000) / 200 = -200 / 200 = -1
The area under the right tail of the normal distribution curve for a z-score of -1 is the same as the area under the left tail for a z-score of 1. By symmetry, we can look up the area in the z-table for a z-score of 1, which is approximately 0.8413.
To determine the percentage, we can transform this value to a percentage and subtract it from 100%:
Percentage = (1 - 0.8413) * 100% = 15.87%
So, approximately 15.87% of the persons who take the test score above 800.
c. What percentage score below 1200?
To find this, we can calculate the z-score for 1200:
z = (1200 - 1000) / 200 = 200 / 200 = 1
The area under the left tail of the normal distribution curve for a z-score of 1 can be found in the z-table and is approximately 0.8413.
To determine the percentage, we can transform this value to a percentage:
Percentage = 0.8413 * 100% = 84.13%
So, approximately 84.13% of the persons who take the test score below 1200.
d. Above what score do 20% of the test-takers score?
Here, we need to find the z-score that corresponds to an area under the right tail of 0.20. Looking at the z-table, the closest value we find is approximately 0.8416.
Now, we need to solve for x in the z-score equation:
0.8416 = (x - 1000) / 200
0.8416 * 200 = x - 1000
x - 1000 = 168.32
x = 1168.32
So, the score above which 20% of the test-takers score is approximately 1168.32.
e. Above what score do 30% of the test-takers score?
Similarly, we need to find the z-score that corresponds to an area under the right tail of 0.30. Looking at the z-table, we find that the closest value is approximately 0.5244.
Now, we can solve for x in the z-score equation:
0.5244 = (x - 1000) / 200
0.5244 * 200 = x - 1000
x - 1000 = 104.88
x = 1104.88
So, the score above which 30% of the test-takers score is approximately 1104.88.