Stats Question

Question

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.

 

Indicate the percentage or score called for by the problem.
 
What percentage of the persons who take the test score above 1300?
What percentage score above 800?
What percentage score below 1200?
Above what score do 20% of the test-takers score?
Above what score do 30% of the test-takers score?
 

Answer 1

Shawanda:

 

Let X be a combined verbal and quantitative score. We are given that the mean is 1000 and the SD is 200.

 

1) What percentage of the persons who take the test score above 1300?

 

We want P[ X > 1300]   This is the area under the curve to the RIGHT.

 

Step 1.  Find the standardized z_score for 1300 by the formula z = (value -mean)/ SD.  Here (z = 1300 -1000)/200 = 1.00.  z_score has a standard normal distribution with mean 0 and SD 1.

 

Step 2.  Find the area under the standard normal curve TO THE LEFT of the z_score either by using the table in your text on (as I am telling you here) using the web-link  

 

http://stattrek.com/online-calculator/normal.aspx

 

Enter 1.00 in the blank box in front of z_score, make sure that the mean and the SD are 0 and 1 resp. Press calculate.  You will see that P(Z ≤ 1.00) = 0.841.  To get the area to the right, subtract  0.841 from 1 to get 0.159.  That is the answer.  Now you do the next similar question.  Post a comment if you get stuck somewhere.

 

3) What percentage score below 1200?  You want P(X < 1200), area to the left of 1200 under the curve.

 

Step 1. Find the standardized z_score as earlier.  We get z = 1.00

 

WE HAVE ALREADY found the required area in Step 2.  Wonderful.  No need to subtract from 1 because we WANT the area to the left.  The answer is 0.841.

 

4) Above what score do 20% of the test-takers score?

 

You want first a z_score (which we will later convert to what we want in our problem) such that P(Z > z_score) = 0.20   Do you see why "above" is translated like this? (above means to the right of the z_score).  If not, post a comment

 

For this question use the web-link

 

http://www.danielsoper.com/statcalc3/calc.aspx?id=19

 

This link gives the "cumulative probability" which is cumulated from below, not above.  But we know that P(Z > z_score) = 0.2 is the same as P(Z ≤ z_score) = 0.8. (The total area is 1)  So to use the link, the cumulative probability is 0.8 (from the left).  Go to the link.  Type in, in front of cumulative probability level,  Get z_score as 0.842.

 

Now we have to convert this z_score to an X score by the same formula but used backwards. That is 0.842 = (X- 1000) / 200.  Solve for X.  Easy.  We get. X = 1168.4.

 

4) Above what score do 30% of the test-takers score?

 

Follow exactly the same stuff as in 3).  If you get stuck, post a comment.

 

"Dr. G."