Exercises

Author(s)

Prerequisites

You may want to use the "Calculate Area for a given X" and the "Calculate X for a given Area" applets for some of these exercises.

1. If scores are normally distributed with a mean of 35 and a standard deviation of 10, what percent of the scores is: (a) greater than 34? (b) smaller than 42? (c) between 28 and 34? (Ch. 6.C)

2. (a) What are the mean and standard deviation of the standard normal distribution? (b) What would be the mean and standard deviation of a distribution created by multiplying the standard normal distribution by 8 and then adding 75? (Ch. 6.E & Ch. 3.D)

3. The normal distribution is defined by two parameters. What are they?
(Ch. 6.A)

4. (a) What proportion of a normal distribution is within one standard deviation of the mean? (b) What proportion is more than 2.0 standard deviations from the mean? (c) What proportion is between 1.25 and 2.1 standard deviations above the mean? (Ch. 6.E)

5. A test is normally distributed with a mean of 70 and a standard deviation of 8. (a) What score would be needed to be in the 85th percentile? (b) What score would be needed to be in the 22nd percentile? (Ch. 6.C)

6. Assume a normal distribution with a mean of 70 and a standard deviation of 12. What limits would include the middle 65% of the cases? (Ch. 6.C)

7. A normal distribution has a mean of 20 and a standard deviation of 4. Find the Z scores for the following numbers: (Ch. 6.E)

(a) 28    (b) 18    (c) 10    (d) 23

8. Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph.

(a) The current speed limit is 65 mph. What is the proportion of vehicles less than or equal to the speed limit?
(b) What proportion of the vehicles would be going less than 50 mph?
(c) A new speed limit will be initiated such that approximately 10% of vehicles will be over the speed limit. What is the new speed limit based on this criterion?
(d) In what way do you think the actual distribution of speeds differs from a normal distribution?
(Ch. 6.C)

9. A variable is normally distributed with a mean of 120 and a standard deviation of 5. One score is randomly sampled. What is the probability it is above 127? (Ch. 6.C)

10. You want to use the normal distribution to approximate the binomial distribution. Explain what you need to do to find the probability of obtaining exactly 7 heads out of 12 flips. (Ch. 6.F)

11. A group pf students at a school takes a history test. The distribution is normal with a mean of 25, and a standard deviation of 4. (a) Everyone who scores in the top 30% of the distribution gets a certificate. What is the lowest score someone can get and still earn a certificate? (b) The top 5% of the scores get to compete in a statewide history contest. What is the lowest score someone can get and still go onto compete with the rest of the state? (Ch. 6.C)

12. Use the normal distribution to approximate the binomial distribution and find the probability of getting 15 to 18 heads out of 25 flips. Compare this to what you get when you calculate the probability using the binomial distribution. Write your answers out to four decimal places. (Ch. 6.F & Ch. 6.G)

13. True/false: For any normal distribution, the mean, median, and mode will have the same value. (Ch. 6.A)

14. True/false: In a normal distribution, 11.5% of scores are greater than Z = 1.2. (Ch. 6.E)

15. True/false: The percentile rank for the mean is 50% for any normal distribution. (Ch. 6.C)

16. True/false: The larger the p, the better the normal distribution approximates the binomial distribution. (Ch. 6.F & Ch. 6.G)

17. True/false: A Z-score represents the number of standard deviations above or below the mean. (Ch. 6.E)

18. True/false: Abraham de Moivre, a consultant to gamblers, discovered the normal distribution when trying to approximate the binomial distribution to make his computations easier. (Ch. 6.B)

19. True/false: The standard deviation of the blue distribution shown below is about 10. (Ch. 6.D)

20. True/false: In the figure below, the red distribution has a larger standard deviation than the blue distribution. (Ch. 6.D)

21. True/false: The red distribution has more area underneath the curve than the blue distribution does. (Ch. 6.A)

Questions from Case Studies:


The following question uses data from the Angry Moods (AM) case study.

22. For this problem, use the Anger Expression (AE) scores. (a) Compute the mean and standard deviation. (b) Then, compute what the 25th, 50th and 75th percentiles would be if the distribution were normal. (c) Compare the estimates to the actual 25th, 50th, and 75th percentiles. (Ch. 6.C)


The following question uses data from the Physicians' Reactions (PR) case study.

23. For this problem, use the time spent with the overweight patients. (a) Compute the mean and standard deviation of this distribution. (b) What is the probability that if you chose an overweight participant at random, the doctor would have spent 31 minutes or longer with this person? (c) Now assume this distribution is normal (and has the same mean and standard devation). Now what is the probability that if you chose an overweight participant at random, the doctor would have spent 31 minutes or longer with this person? (Ch. 6.C)

Answers:

1) (b) 75.8%

2) (b) Mean = 75

4) (c) .088

5) (a) 78.3

7) (a) 2.0

8) (a) .227

11) (a) 27.1

12) .2037 (normal approximation)

22) 25th percentile: (b) 28.27 (c) 26.75

23) (b) .053

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