Chapter 3 Exercises

David M. Lane

Prerequisites

All material presented in chapter 3

1. Make up a dataset of 12 numbers with a positive skew. Use a statistical program to compute the skew. Is the mean larger than the median as it usually is for distributions with a positive skew? What is the value for skew? (Ch. 3.C & Ch 3.A.3)

2. Repeat Problem 3 only this time make the dataset have a negative skew. (Ch. 3.C & Ch 3.A.3)

3. Make up three data sets with 5 numbers each that have:
(a) the same mean but different standard deviations.
(b) the same mean but different medians.
(c) the same median but different means.
(Ch. 3.A.2.a & Ch 3.B.1)

4. Find the mean and median for the following three variables:
(Ch. 3.A.2.a)

A B C
8 4 6
5 4 2
7 6 3
1 3 4
3 4 1

5. A sample of 30 distance scores measured in yards has a mean of 7, a variance of 16, and a standard deviation of 4. (a) You want to convert all your distances from yards to feet, so you multiply each score in the sample by 3. What are the new mean, variance, and standard deviation? (b) You then decide that you only want to look at the distance past a certain point. Thus, after multiplying the original scores by 3, you decide to subtract 4 feet from each of the scores. Now what are the new mean, variance, and standard deviation? (Ch. 3.D)

6. You recorded the time in seconds it took for 8 participants to solve a puzzle. These times appear below. However, when the data was entered into the statistical program, the score that was supposed to be 22.1 was entered as 21.2. You had calculated the following measures of central tendency: the mean, the median, and the mean trimmed 25%. Which of these measures of central tendency will change when you correct the recording error? (Ch. 3.A.2.a & Ch 3.A.2.g)

15.2
18.8
19.3
19.7
20.2
21.8
22.1
29.4

7. For the test scores in question #6, which measures of variability (range, standard deviation, variance) would be changed if the 22.1 data point had been erroneously recorded as 21.2? (Ch 3.B.1)

8. You know the minimum, the maximum, and the 25th, 50th, and 75th percentiles of a distribution. Which of the following measures of central tendency or variability can you determine?
(Ch. 3.A.2.a, Ch 3.A.2.g & Ch 3.B.1)

mean, median, mode, trimean, geometric mean,
range, interquartile range, variance, standard deviation

9. For the numbers 1, 3, 4, 6, and 12: (a) Find the value (v) for which S(X-v)2 is minimized. (b) Find the value (v) for which S |x-v| is minimized. (Ch. 3.A.2.e)

10. Your younger brother comes home one day after taking a science test. He says that someone at school told him that "60% of the students in the class scored above the median test grade." What is wrong with this statement? (Ch. 3.A.2.a)

11. An experiment compared the ability of three groups of participants to remember briefly-presented chess positions. The data are shown below. The numbers represent the number of pieces correctly remembered from three chess positions. Compare the performance of each group. Consider spread as well as central tendency. (Ch. 3.A.2.a, Ch 3.A.2.g & Ch 3.B.1)

 Non-players Beginners Tournament players 22.1 32.5 40.1 22.3 37.1 45.6 26.2 39.1 51.2 29.6 40.5 56.4 31.7 45.5 58.1 33.5 51.3 71.1 38.9 52.6 74.9 39.7 55.7 75.9 43.2 55.9 80.3 43.2 57.7 85.3

12. True/False: A bimodal distribution has two modes and two medians. (Ch. 3.A.2.a)

13. True/False: The best way to describe a skewed distribution is to report the mean. (Ch. 3.A.3)

14. True/False: When plotted on the same graph, a distribution with a mean of 50 and a standard deviation of 10 will look more spread out than will a distribution with a mean of 60 and a standard deviation of 5. (Ch 3.B.1)

Questions from Case Studies:

The following questions are from the Angry Moods (AM) case study.

15. (AM#4) Does Anger-Out have a positive skew, a negative skew, or no skew? (Ch. 3.C)

16. (AM#8) What is the range of the Anger-In scores? What is the interquartile range? (Ch 3.B.1)

17. (AM#12) What is the overall mean Control-Out score? What is the mean Control-Out score for the athletes? What is the mean Control-Out score for the non-athletes? (Ch. 3.A.2.a)

18. (AM#15) What is the variance of the Control-In scores for the athletes? What is the variance of the Control-In scores for the non-athletes? (Ch 3.B.1)

The following question is from the Flatulence (F) case study.

19. (F#2) Based on a histogram of the variable "perday", do you think the mean or median of this variable is larger? Calculate the mean and median to see if you are right. (Ch. 3.A.3 & Ch. 2.C.2)

The following questions are from the Stroop (S) case study.

20. (S#1) Compute the mean for "words". (Ch. 3.A.2.a)

21. (S#2) Compute the mean and standard deviation for "colors".
(Ch. 3.A.2.a & Ch 3.B.1)

The following questions are from the Physicians' Reactions (PR) case study.

22. (PR#2) What is the mean expected time spent for the average-weight patients? What is the mean expected time spent for the overweight patients? (Ch. 3.A.2.a)

23. (PR#3) What is the difference in means between the groups? By approximately how many standard deviations do the means differ?
(Ch. 3.A.2.a & Ch 3.B.1)

The following question is from the Smiles and Leniency (SL) case study.

24. (SL#2) Find the mean, median, standard deviation, and interquartile range for the leniency scores of each of the four groups. (Ch. 3.A.2.a & Ch 3.B.1)

The following questions are from the ADHD Treatment (AT) case study.

25. (AT#4) What is the mean number of correct responses of the participants after taking the placebo (0 mg/kg)? (Ch. 3.A.2.a)

26. (AT#7) What are the standard deviation and the interquartile range of the d0 condition? (Ch 3.B.1)

4) Variable A: Mean = 4.8, Median = 5

5) (a) Mean = 21, Var = 144, SD = 12

9) (a) 5.2

16) Range = 21

17) Non-athletes: 23.2

18) Athletes: 20.5

21) Mean = 20.2

22) Ave. weight: 31.4

24) False smile group:
Mean = 5.37
Median = 5.50
SD = 1.83
IQR = 3.0

26) SD = 11.3