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# Sample Worksheet: Normal Distributions -- GRE, GMAT, Statistics

Many of my students preparing for the GRE or GMAT have decent algebraic skills, but most have trouble with statistical reasoning --- for a variety of reasons. Some have never had statistics; others have been away from it for years. In either case, it's crucial to get up to speed on the basics!

To get a sense of how prepared you are for some of the more challenging statistics questions on the GRE and GMAT, check out the following worksheet I've developed. When you work with me, you'll gain exactly the skills you need to ace these and similar problems --- you'll learn to complete this entire sheet in fewer than five minutes, in fact! --- and have access to a wide range of specialized materials I've developed over the years, materials full of strategies and problem sets you won't find in any published prep book. I guarantee you won't find a more helpful or expert tutor, so send me an email today!

PROBLEM SET: NORMAL DISTRIBUTIONS

Exercises: Percentiles & The Empirical Rule

Assume the variable X is normally distributed with mean m and standard deviation s. Fill in the blanks to make the statements true:

1. _____ percent of the data for X lies within one standard deviation of m.

2. _____ percent of the data for X lies between m and m+2s.

3. _____ percent of the data for X lies further than one standard deviation from m.

4. _____ percent of the data for X is less than m and greater than m+s.

5. _____ percent of the data for X lies between m-2s and m-s.

6. _____ percent of the data for X lies within three standard deviations of m.

7. _____ percent of the data for X lies further than two standard deviations from m.

8. _____ percent of the data for X lies above m+3s.

9. _____ percent of the data for X lies below m-s.

10. ____ percent of the data for X lies between m-2s and m+s.

11. If x is at the 2.5th percentile, then x is ____ standard deviations from the mean.

12. If x is at the 99th percentile, then x is more than ____ standard deviations from the mean. (Choose the greatest integer that makes this statement true.)

13. If x is at the 20th percentile, then x is between ____ & ____ standard deviations from the mean. (Choose the tightest bounds.)

14. If x is at the 51st percentile, then x is between ____ & ____ standard deviations from the mean. (Choose the tightest bounds.)

15. If x is at the 38th percentile, then x is between ____ & ____ standard deviations from the mean. (Choose the tightest bounds.)

16. If x is two standard deviations above the mean, then x is at the ______ percentile.

17. If x is one standard deviation below the mean, then x is at the ______ percentile.

18. If x is two standard deviations below the mean, then x is at the ______ percentile.

19. If x is one standard deviation above the mean, then x is at the ______ percentile.

20. If x lies between m-2s and m+3s, then x lies between the ______ and ______ percentiles. (Choose the tightest bounds.)

Problems

These problems are harder than the routine exercises above; you'll have to be more creative to answer them.

1. (Quantitative Comparison) Suppose that the variable X is normally distributed with a mean of 250 and that 70% of the data lies between 240 and 270.

Column A: the standard deviation of the variable X

Column B: 10

2. (Quantitative Comparison) Suppose that the variable X is normally distributed with mean 50 and standard deviation 10.

Column A: the percentage of data in X that lies between 30 and 35

Column B: the percentage of data in X that lies between 35 and 40

HINT: Think about the shape of the normal distribution!

3. (Quantitative Comparison) Suppose that the variable X is normally distributed with mean 50 and standard deviation 10.

Column A: the percentage of data in X that lies between 35 and 40

Column B: 7

HINT: Think about what you learned from Problem 2!

4. (Multiple Answer Multiple Choice) Researchers collected 400 observations of GRE verbal scores and found them to be approximately normally distributed. Suppose Jane scored exactly two standard deviations above the mean, while John, sadly, scored only at the 5th percentile with a verbal score of 350.

Based on this information only, which of the following must be true? (Select all that apply.)

A. Fewer than 380 people scored higher than John but lower than Jane.

B. More than 8 people earned a score higher than Jane’s.

C. Jane scored closer to the mean than John did.

D. More than one person earned the same score as John.

E. Jane scored higher than a 600.

5. Suppose that X and Y both measure IQ, but in distinct populations, and that X and Y are both approximately normally distributed, each with a mean of 100. The standard deviation of X is 15, however, while the standard deviation of Y is 5.

Part I (Quantitative Comparison)

Column A: the probability a person randomly chosen from population X has an IQ between 70 and 115

Column B: the probability a person randomly chosen from population Y has an IQ between 75 and 100

Part II (Quantitative Comparison) A person is randomly chosen from one of the two populations and found to have an IQ of 140.

Column A: the probability that the person was randomly chosen from population X

Column B: the probability that the person was randomly chosen from population Y