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. About what score do 20% of the test-takers score?
e. About what score do 30% of the test-takers score?

a, b, c. Z = (score-mean)/SD

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.

d, e. Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability and its Z score. Insert Z value into above equation and solve for the score. However, you need to designate whether you are talking about higher, lower or ordinate percentage.

To answer these questions, we need to use the properties of the normal distribution curve. We'll calculate each of the percentages based on the given mean, standard deviation, and the range of scores.

(a) To find the percentage of persons who score above 1300 on the GRE, we first need to calculate the z-score for this score. The z-score formula is given by:

z = (x - mean) / standard deviation,

where x is the score we're interested in. Therefore, for x = 1300:

z = (1300 - 1000) / 200 = 3.

Next, we can use a z-table or a calculator to look up the percentage that corresponds to a z-score of 3. From the z-table, we find that this percentage is approximately 0.0013, or 0.13%.

So, the percentage of persons who score above 1300 is around 0.13%.

(b) Similar to part (a), we'll calculate the z-score for a score of 800:

z = (800 - 1000) / 200 = -1.

Again, using the z-table, we find that the percentage corresponding to a z-score of -1 is approximately 0.1587, or 15.87%.

Therefore, around 15.87% of persons score above 800.

(c) To find the percentage of persons who score below 1200, we can subtract the percentage above 1200 from 100%. Since we already calculated the percentage above 1300 in part (a), we can use the same z-score of 3 and subtract it from 100%:

100% - 0.13% = 99.87%.

So, around 99.87% of persons score below 1200.

(d) To find the score at which 20% of the test-takers score, we need to find the z-score that corresponds to the 20th percentile. This can be obtained from the z-table, and the closest value is approximately -0.84.

Using the z-score formula, we can solve for x:

-0.84 = (x - 1000) / 200.

Solving for x, we get:

-0.84 * 200 + 1000 = x,

x ≈ 832.

Therefore, around 20% of the test-takers score around 832.

(e) Similarly, to find the score at which 30% of the test-takers score, we need to find the z-score that corresponds to the 30th percentile. Using the z-table, the closest value is approximately -0.52.

Again, using the z-score formula and solving for x:

-0.52 = (x - 1000) / 200.

Solving for x, we get:

-0.52 * 200 + 1000 = x,

x ≈ 896.

So, approximately 30% of the test-takers score around 896.