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?

I would like to have the correct answer, please.

To solve these problems, we need to use the properties of the normal distribution curve.

(a) To find the percentage of persons who score above 1300:
1. Start by drawing a normal distribution curve representing the GRE scores.
2. Mark the mean (1000) on the horizontal axis.
3. Calculate the Z-score for 1300 using the formula Z = (X - mean) / standard deviation. In this case, X = 1300, mean = 1000, and standard deviation = 200. The Z-score is (1300 - 1000) / 200 = 1.5.
4. Shade the area under the curve to the right of the Z-score of 1.5.
5. Consult a Z-table or use a statistical calculator to find the percentage of the area shaded. In this case, the percentage above 1300 is approximately 6.68%.

(b) To find the percentage of persons who score above 800:
1. Draw the normal distribution curve as before.
2. Calculate the Z-score for 800 using the formula Z = (X - mean) / standard deviation. In this case, X = 800, mean = 1000, and standard deviation = 200. The Z-score is (800 - 1000) / 200 = -1.
3. Shade the area under the curve to the right of the Z-score of -1.
4. Find the percentage of the area shaded using a Z-table or statistical calculator. The percentage above 800 is approximately 84.13%.

(c) To find the percentage who score below 1200:
1. Draw the normal distribution curve.
2. Calculate the Z-score for 1200 using the formula Z = (X - mean) / standard deviation. In this case, X = 1200, mean = 1000, and standard deviation = 200. The Z-score is (1200 - 1000) / 200 = 1.
3. Shade the area under the curve to the left of the Z-score of 1.
4. Find the percentage of the shaded area using a Z-table or statistical calculator. The percentage below 1200 is approximately 84.13%.

(d) To find the score above which 20% of test-takers score:
1. Draw the normal distribution curve.
2. Shade the area under the curve to the right such that the percentage shaded is 20% (0.20).
3. Find the corresponding Z-score using a Z-table or statistical calculator for the shaded area of 0.20.
4. Use the Z-score formula Z = (X - mean) / standard deviation, and solve for X by substituting the Z-score and known values of mean (1000) and standard deviation (200). X = Z * standard deviation + mean. In this case, Z ≈ 0.84. So, X = 0.84 * 200 + 1000 = 1168.

(e) To find the score above which 30% of test-takers score:
1. Draw the normal distribution curve.
2. Shade the area under the curve to the right such that the percentage shaded is 30% (0.30).
3. Find the corresponding Z-score using a Z-table or statistical calculator for the shaded area of 0.30.
4. Use the Z-score formula Z = (X - mean) / standard deviation, and solve for X by substituting the Z-score and known values of mean (1000) and standard deviation (200). X = Z * standard deviation + mean. In this case, Z ≈ 0.52. So, X = 0.52 * 200 + 1000 = 1104.

Remember, the values obtained for (d) and (e) are approximate since we are rounding Z-scores.