The Graduate Record Exam 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.
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?
I thought I answered this previously.
a-c. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability of each Z score. Multiply by 100.
d, e. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.20 and .30 respectively) and get each Z score. Insert into equation above to calculate each raw score.
To solve these problems, we need to convert the scores into Z-scores, which are a measure of how many standard deviations a particular score is from the mean. The Z-score formula is:
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the raw score
μ is the mean
σ is the standard deviation
Let's calculate the Z-scores for each problem:
A. To find the percentage of test takers who score above 1300, we need to calculate the Z-score for 1300 and find the area to the right of that Z-score.
Z = (1300 - 1000) / 200 = 3
From the Z-table, we find that the area to the right of Z = 3 is approximately 0.0013. Therefore, approximately 0.13% of test takers score above 1300.
B. To find the percentage of test takers who score above 800, we follow the same steps:
Z = (800 - 1000) / 200 = -1
From the Z-table, we find that the area to the right of Z = -1 is approximately 0.8413. However, we want the percentage above 800, so we subtract it from 1. Therefore, approximately 1 - 0.8413 = 0.1587, or 15.87% of test takers score above 800.
C. To find the percentage of test takers who score below 1200, we calculate the Z-score and find the area to the left of that Z-score.
Z = (1200 - 1000) / 200 = 1
From the Z-table, we find that the area to the left of Z = 1 is approximately 0.8413. Therefore, approximately 84.13% of test takers score below 1200.
D. To find the score above which 20% of test takers score, we look for the Z-score that corresponds to the area to the left of 0.2.
From the Z-table, we find that the Z-score corresponding to an area of 0.2 to the left is approximately -0.84. Now, we can solve for X:
-0.84 = (X - 1000) / 200
Solving for X, we get:
X = -0.84 * 200 + 1000 = 831.2
Therefore, 20% of the test takers score above approximately 831.2.
E. To find the score above which 30% of test takers score, we look for the Z-score that corresponds to the area to the left of 0.3.
From the Z-table, we find that the Z-score corresponding to an area of 0.3 to the left is approximately -0.52. Now, we can solve for X:
-0.52 = (X - 1000) / 200
Solving for X, we get:
X = -0.52 * 200 + 1000 = 896
Therefore, 30% of the test takers score above approximately 896.