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1. Across the USA, results for these exams are normally distributed. What does that mean

and why is this the case?

This means that the plot of the frequency of exam scores resembles a normal distribution. This is

the case because there is a large number of people taking the exam. Thus the Central Limit

Theorem applies and asserts that the distribution of the results should tend toward the mean.

Also, the exam is designed to elicit a normal distribution. The exam writers do not choose one

small portion of the range of topics for the exam and ask detailed questions about topics within

that range. This would give a bimodal or possibly skewed distribution.

2. If you were to create a histogram of all GRE scores, what would you expect the

histogram to look like? Would it be symmetrical? Would it be bell shaped? How many

modes would it likely have? Would it be skewed?

I would expect the histogram to by symmetrical with the shape of an inverted bell. It would

likely have 1 mode located within 1 standard deviation of the mean. It could be skewed. The

distribution would be centered about the mean, and the fact that few students earn the lowest

score while few but several students earn the maximum score implies that it would probably be

skewed right, if at all.

3. Suppose that the mean GRE score for the USA is 500 and the standard deviation is 75.

Use the Empirical Rule (also called the 68-95-99.7 Rule) to determine the percentage of

students likely to get a score between 350 and 650? What percentage of students will get

a score above 500? What percentage of students will get a score below 275? Is a score

below 275 significantly different from the mean? Why or why not?

500 – 350 = 150 = 2⋅75, and 650 – 500 = 150 = 2⋅75. Thus each score is 2 standard deviations

from the mean. By the Empirical Rule, 95% of students are likely to score within 2 standard

deviations of the mean.

Since the mean is 50 and half of the scores lie above the mean while half lie below, 50% of

students will get a score above 500.

500 – 275 = 225 = 3⋅75. Because we are only concerned about the portion of the curve to the left

of 3 standard deviations to the left of the mean, we subtract 1 - .997 = .003 to get the area under

both tails of the curve past 3 standard deviations. We only want the area under 1 tail, so divide

by 2 to get .003/2 = .0015.

A score below 275 is significantly different from the mean because it lies outside the standard

95% confidence interval from the mean. It even lies outside the 99% confidence interval.

4. Choose any GRE score between 200 and 800. Be sure that you do not choose a score that

a fellow student has already selected. Using your chosen score, how many standard

deviations from the mean is your score? (This value is called the z-value). Using the table

above (or the z table in Doc Sharing), what percentage of students will likely get a score

below this value? What percentage of students are likely to get a score above this value?

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1. Across the USA, results for these exams are normally distributed. What does that mean and why is this the case? This means that the plot of the frequency of exam scores resembles a normal distribution. This is the case because there is a large number of people taking the exam. Thus the Central Limit Theorem applies and asserts that the distribution of the results should tend toward the mean. Also, the exam is designed to elicit a normal distribution. The exam writers do not choose one small portion of the range of topics for the exam and ask detailed questions about topics within that range ...
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