The Graduate Record Examination (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:
1. Draw a rough sketch, darkening the portion of the cure that relates to the answer,
2. Indicate the percentage or score called for by the problem.
a. What percentage of 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 about 20 percent of the test-takers score?
e. Above what score do about 30 percent of the test-takers score
Cannot draw on these posts.
Z = (score-mean)/SD
a-c. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores. Multiply by 100.
d-e. Use same table for .20 and .30 to get Z scores. Insert in equation above to get raw score.
To solve these problems, we will need to use the normal distribution curve. The mean of the scores is given as 1000, with a standard deviation of 200. The scores range from 200 to 1600.
To draw a rough sketch, we will create a normal distribution curve with the mean at 1000 and the standard deviation of 200. The scores range from 200 to 1600, so we will make sure the curve covers this range. We will darken the portion of the curve that relates to the answer.
a. What percentage of persons who take the test score above 1300?
To find the percentage of people who score above 1300, we need to determine the area under the curve to the right of 1300.
First, let's calculate the z-score for 1300 using the formula:
z = (x - μ) / σ
where x is the score we want the z-score for, μ is the mean, and σ is the standard deviation.
z = (1300 - 1000) / 200 = 3/2 = 1.5
Looking up the corresponding percentage in the z-table, we find that the area to the right of 1.5 is approximately 0.0668, or 6.68%.
So, the percentage of persons who score above 1300 is approximately 6.68%.
b. What percentage score above 800?
Similarly, to find the percentage of people who score above 800, we need to determine the area under the curve to the right of 800.
Calculating the z-score for 800:
z = (800 - 1000) / 200 = -1
Looking up the corresponding percentage in the z-table, we find that the area to the right of -1 is approximately 0.8413, or 84.13%.
Therefore, the percentage of persons who score above 800 is approximately 84.13%.
c. What percentage score below 1200?
To find the percentage of people who score below 1200, we need to determine the area under the curve to the left of 1200.
Calculating the z-score for 1200:
z = (1200 - 1000) / 200 = 1
Looking up the corresponding percentage in the z-table, we find that the area to the left of 1 is approximately 0.8413, or 84.13%.
Therefore, the percentage of persons who score below 1200 is approximately 84.13%.
d. Above what score do about 20 percent of the test-takers score?
To find the score above which about 20 percent of the test-takers score, we need to find the z-score corresponding to that percentile.
Using the z-table, we find that a percentile of 20% corresponds to a z-score of approximately -0.8416.
Calculating the score:
-0.8416 = (x - 1000) / 200
x - 1000 = -0.8416 * 200
x - 1000 = -168.32
x = -168.32 + 1000
x = 831.68
Therefore, approximately 20 percent of the test-takers score above a score of 831.68.
e. Above what score do about 30 percent of the test-takers score?
Using the z-table, we find that a percentile of 30% corresponds to a z-score of approximately -0.5244.
Calculating the score:
-0.5244 = (x - 1000) / 200
x - 1000 = -0.5244 * 200
x - 1000 = -104.88
x = -104.88 + 1000
x = 895.12
Therefore, approximately 30 percent of the test-takers score above a score of 895.12.
To solve these problems, we need to use z-scores and the standard normal distribution table. The formula for calculating the z-score is:
z = (x - mean) / standard deviation
where x is the raw score, mean is the mean score, and standard deviation is the standard deviation of the scores.
Let's solve each problem step by step:
a. What percentage of persons who take the test score above 1300?
To find the percentage of test-takers who score above 1300, we first need to calculate the z-score for 1300 using the formula:
z = (1300 - mean) / standard deviation
z = (1300 - 1000) / 200
z = 3
From the standard normal distribution table, we can find that the area to the left of z = 3 is 0.9987. Since we want the percentage above 1300, we subtract this value from 1:
Percentage above 1300 = 1 - 0.9987 = 0.0013
b. What percentage score above 800?
Similarly, we need to calculate the z-score for 800:
z = (800 - 1000) / 200
z = -1
From the standard normal distribution table, we can find that the area to the left of z = -1 is 0.1587. Again, we subtract this value from 1 to find the percentage above 800:
Percentage above 800 = 1 - 0.1587 = 0.8413
c. What percentage score below 1200?
We need to calculate the z-score for 1200:
z = (1200 - 1000) / 200
z = 1
From the standard normal distribution table, the area to the left of z = 1 is 0.8413.
Percentage below 1200 = 0.8413
d. Above what score do about 20 percent of the test-takers score?
To find the score above which about 20 percent of test-takers score, we look for the z-score that corresponds to an area of 0.2 in the standard normal distribution table.
From the table, we find that the z-score corresponding to an area of 0.2 is approximately -0.84.
Now, we can solve for the raw score (x) using the z-score formula:
-0.84 = (x - 1000) / 200
Simplifying the equation, we get:
x - 1000 = -0.84 * 200
x - 1000 = -168
x = 1000 - 168
x = 832
About 20 percent of test-takers score above 832.
e. Above what score do about 30 percent of the test-takers score?
Similar to the previous problem, we need to find the z-score corresponding to an area of 0.3.
From the standard normal distribution table, the z-score corresponding to an area of 0.3 is approximately -0.52.
Using the z-score formula, we have:
-0.52 = (x - 1000) / 200
Simplifying the equation, we get:
x - 1000 = -0.52 * 200
x - 1000 = -104
x = 1000 - 104
x = 896
About 30 percent of test-takers score above 896.
Drawing a rough sketch with the corresponding shaded areas will help visualize the distribution and the portions relating to the answers.