The verbal part of the Graduate Record Exam (GRE) has a mean of 500 and a standard deviation of 100. Use the normal distribution to answer the following questions:

a. If you wanted to select only student at or above the 90th percentile, what verbal GRE score would you use as a cutoff score?
b. What verbal GRE score corresponds to a percentile rank of 15%? What verbal GRE score corresponds to a percentile rank of 55%?
c. What's the percentile rank of a GRE score of 628? What's the percentile rank of a GRE score of 350?
d. If you randomly selected 1,500 students who had taken the verbal GRE, how many would you expect to score lower than 250? How many would you expect to score higher than 750?

4. The verbal part of the Graduate Record Exam (GRE) has a ƒÝ of 500 and ƒã = 100. Use the normal distribution to answer the following questions:

a. If you wanted to select only students at or above the 90th percentile, what verbal GRE score would you use as a cutoff score?
b. What verbal GRE score corresponds to a percentile rank of 15%? What verbal GRE score corresponds to a percentile rank of 55%?
c. What¡¦s the percentile rank of a GRE score of 628? What¡¦s the percentile rank of a GRE score of 350?
d. If you randomly selected 1,500 students who had taken the verbal GRE, how many would you expect to score lower than 250? How many would you expect to score higher than 750?

a. To find the verbal GRE score used as a cutoff for the 90th percentile, we need to find the z-score corresponding to this percentile and then convert it back to the original scale.

1. First, we need to find the z-score that corresponds to the 90th percentile. Since the distribution is normal, we know that the z-score corresponding to the 90th percentile is roughly 1.28.

2. Using the formula for z-score, we can convert this z-score back to the original scale:
z = (x - mean) / standard deviation

Rearranging the formula, we get:
x = z * standard deviation + mean

Plugging in the values, we have:
x = 1.28 * 100 + 500 = 628

Therefore, the verbal GRE score used as a cutoff for the 90th percentile would be 628.

b. To find the verbal GRE score corresponding to a given percentile rank, we need to find the z-score corresponding to that rank and then convert it back to the original scale.

1. For a percentile rank of 15%, we need to find the z-score that corresponds to this rank. Using a z-table or calculator, we find that the z-score corresponding to the 15th percentile is roughly -1.04.

2. Converting the z-score back to the original scale:
x = z * standard deviation + mean
x = -1.04 * 100 + 500 = 396

Therefore, the verbal GRE score corresponding to a percentile rank of 15% is 396.

For a percentile rank of 55%, we repeat the same process:
1. Using a z-table or calculator, we find that the z-score corresponding to the 55th percentile is roughly 0.125.

2. Converting the z-score back to the original scale:
x = 0.125 * 100 + 500 = 512.5

Rounded to the nearest whole number, the verbal GRE score corresponding to a percentile rank of 55% is 513.

c. To find the percentile rank of a given GRE score, we need to find the z-score first and then convert it to a percentile.

1. For a GRE score of 628:
z = (x - mean) / standard deviation
z = (628 - 500) / 100 = 1.28

Using a z-table or calculator, we find that the percentile corresponding to a z-score of 1.28 is approximately 90th percentile.

Therefore, a GRE score of 628 is at the 90th percentile.

2. For a GRE score of 350:
z = (x - mean) / standard deviation
z = (350 - 500) / 100 = -1.5

Using a z-table or calculator, we find that the percentile corresponding to a z-score of -1.5 is approximately 6.68th percentile.

Therefore, a GRE score of 350 is at the 6.68th percentile.

d. To find the expected number of students scoring lower or higher than a certain score, we need to find the proportion of students below or above that score using the z-score and then multiply it by the total number of students.

1. For a score of 250:
z = (x - mean) / standard deviation
z = (250 - 500) / 100 = -2.5

Using a z-table or calculator, we find that the proportion of students scoring lower than a z-score of -2.5 is approximately 0.00621.

Expected number of students scoring lower than 250 = 0.00621 * 1500 = 9.32
Rounded to the nearest whole number, we would expect around 9 students to score lower than 250.

2. For a score of 750:
z = (x - mean) / standard deviation
z = (750 - 500) / 100 = 2.5

Using a z-table or calculator, we find that the proportion of students scoring higher than a z-score of 2.5 is approximately 0.99379.

Expected number of students scoring higher than 750 = 0.99379 * 1500 = 1490.69
Rounded to the nearest whole number, we would expect around 1491 students to score higher than 750.

a. To find the verbal GRE score that corresponds to the 90th percentile, we need to use the Z-score formula:

Z = (X - μ) / σ

Where:
- X is the score we want to find,
- μ is the mean (500), and
- σ is the standard deviation (100).

To find the Z-score corresponding to the 90th percentile, we can use the Z-table or a statistical calculator. The Z-score for the 90th percentile is approximately 1.28.

Plugging this value into the Z-score formula:
1.28 = (X - 500) / 100

Solving for X:
X - 500 = 1.28 * 100
X - 500 = 128
X = 500 + 128
X = 628

Therefore, the cutoff score for selecting only students at or above the 90th percentile is 628.

b. To find the verbal GRE score corresponding to a percentile rank of 15% and 55%, we again use the Z-score formula.

For a percentile rank of 15%:
Z = -1.04 (approximately)

Plugging this Z-score into the formula:
-1.04 = (X - 500) / 100

Solving for X:
X - 500 = -1.04 * 100
X - 500 = -104
X = 500 - 104
X = 396

Therefore, a verbal GRE score of 396 corresponds to a percentile rank of 15%.

For a percentile rank of 55%:
Z = 0.125 (approximately)

Plugging this Z-score into the formula:
0.125 = (X - 500) / 100

Solving for X:
X - 500 = 0.125 * 100
X - 500 = 12.5
X = 500 + 12.5
X = 512.5

Therefore, a verbal GRE score of 512.5 corresponds to a percentile rank of 55%.

c. To find the percentile rank of a given GRE score, we can again use the Z-score formula.

For a GRE score of 628:
Z = (628 - 500) / 100
Z = 1.28 (approx.)

Using the Z-table or a statistical calculator, we can find that the percentile rank for a Z-score of 1.28 is approximately 90%.

Therefore, a GRE score of 628 corresponds to a percentile rank of approximately 90%.

For a GRE score of 350:
Z = (350 - 500) / 100
Z = -1.5 (approx.)

Using the Z-table or a statistical calculator, we can find that the percentile rank for a Z-score of -1.5 is approximately 6.7%.

Therefore, a GRE score of 350 corresponds to a percentile rank of approximately 6.7%.

d. To find the expected number of students scoring lower than 250 or higher than 750, we need to use the formula for cumulative probability:

P(Z < -2.5) = 0.0062 (approx.)
P(Z > 2.5) = 0.0062 (approx.)

The cumulative probability is the probability of scoring below or above a certain Z-score, which corresponds to a specific score on the GRE.

To find the expected number of students, we multiply the probability by the total number of students:

Number of students scoring lower than 250 = 0.0062 * 1,500 ≈ 9.3 (approximately)

Number of students scoring higher than 750 = 0.0062 * 1,500 ≈ 9.3 (approximately)

Therefore, we would expect approximately 9 students to score lower than 250 and approximately 9 students to score higher than 750 out of the randomly selected 1,500 students.

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(.90, .15, etc.) and related Z score. Insert into above equation and solve for score.

For last two (c & d) solve for Z and use the same table.