Each year, thousands of college seniors take the Graduate Record Examination (GRE). Assume the scores are transformed so they have a mean of 600 and a standard deviation of 150. Furthermore, the scores are known to be normally distributed. Determine the percentage of students that score 475 or greater, below 750, and either less than 750 or more than 800?

780

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to those Z scores.

For either-or probabilities, add the individual probabilities.

To determine the percentage of students that score 475 or greater, below 750, and either less than 750 or more than 800, we need to use the concept of the standard normal distribution.

Step 1: Finding the z-score for each score range:
To find the z-score for a specific value, we use the formula:
z = (X - μ) / σ
Where:
- X is the given value,
- μ is the mean of the distribution, and
- σ is the standard deviation of the distribution.

For 475 or greater:
z₁ = (475 - 600) / 150 = -0.83

For below 750:
z₂ = (750 - 600) / 150 = 1.00

For less than 750 or more than 800:
z₃ (less than 750) = (750 - 600) / 150 = 1.00
z₄ (more than 800) = (800 - 600) / 150 = 1.33

Step 2: Finding the percentage using the standard normal distribution table:
Once we have the z-scores, we can use the standard normal distribution table (also known as the z-table or cumulative distribution table) to find the percentage.

For 475 or greater:
Looking up the z-score -0.83 in the z-table, we find the corresponding percentage as 0.2033. This means that approximately 20.33% of students score 475 or greater.

For below 750:
Looking up the z-score 1.00 in the z-table, we find the corresponding percentage as 0.8413. This means that approximately 84.13% of students score below 750.

For less than 750 or more than 800:
To find the percentage of either less than 750 or more than 800, we need to subtract the percentage of scores between 750 and 800 from 100% (as it is the complement).

Looking up the z-score 1.00 in the z-table, we find the corresponding percentage as 0.8413.
Looking up the z-score 1.33 in the z-table, we find the corresponding percentage as 0.9088.

So, the percentage of either less than 750 or more than 800 is approximately:
(1 - 0.8413) + (1 - 0.9088) = 0.1587 + 0.0912 = 0.2499 or 24.99%.

In summary:
- Percentage of students scoring 475 or greater: approximately 20.33%.
- Percentage of students scoring below 750: approximately 84.13%.
- Percentage of students scoring either less than 750 or more than 800: approximately 24.99%.