Student Manual for Fundamental Statistics for the Behavioral Sciences (7th edition)
David C. Howell
The University of Vermont
Contents
Chapter 1 Introduction
Chapter 2 Basic Concepts
Chapter 3 Displaying Data
Chapter 4 Measures of Central Tendency
Chapter 5 Measures of Variability
Chapter 6 The Normal Distribution
Chapter 7 Basic Concepts of Probability
Chapter 8 Sampling Distributions and Hypothesis Testing
Chapter 9 Correlation
Chapter 10 Regression
Chapter 11 Multiple Regression
Chapter 12 Hypothesis Tests Applied to Means: One Sample
Chapter 13 Hypothesis Tests Applied to Means: Two Related Samples
Chapter 14 Hypothesis Tests Applied to Means: Two Independent Samples
Chapter 15 Power
Chapter 16 One-way Analysis of Variance
Chapter 17 Factorial Analysis of Variance
Chapter 18 Repeated-Measures Analysis of Variance
Chapter 19 Chi-Square
Chapter 20 Nonparametric and Distribution-Free Statistical Tests
Chapter 21 Choosing the Appropriate Analysis
Preface
The purpose of this manual is to provide answers to students using the accompanying text, Fundamental Statistics for the Behavioral Sciences, 7th ed. I have provided complete answers to all of the odd-numbered questions. I am often asked for answers to even-numbered exercises as well. I do not provide those because many instructors want to have exercises without answers. I am attempting to balance the two competing needs.
You may find on occasion that you do not have the same answer that I do. Much of this will depend on the degree to which you or I round off intermediate steps. Sometimes it will make a surprising difference. If your answer looks close to mine, and you did it the same way that I did, then don’t worry about small differences. It is even possible that I made an error.
I know that there will be errors in some of these answers. There always are. Even the most compulsive problem solver is bound to make errors, and it has been a long time since anyone accused me of being compulsive. I do try, honest I do, but something always slips past—sometimes they even slip past while I am correcting another error. So I maintain a page on the web listing the errors that I and other have found. If you find an error (minor and obvious typos don’t count unless they involve numbers), please check there and let me know if it is a new one. Some classes even compete to see who can find the most errors—it’s rough when you have to compete with a whole class.
The address for the main web page, is
http://www.uvm.edu/~dhowell/fundamentals/ , and the link to the Errata is there.
Important note: Due to the way hypertext links are shown by Microsoft Word, the underlining often obscures a single underline character, as in “More_Stuff.” If you see a space in an address, it is often really a “_.”
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Chapter 1-Introduction
1.1 A good example is the development of tolerance to caffeine. People who do not normally drink caffeinated coffee are often startled by the effect of one or two cups of regular coffee, whereas those who normally drink regular coffee see no such effect. To test for a context effect of caffeine, you would first need to develop a dependent variable measuring the alerting effect of caffeine, which could be a vigilance task. You could test for a context effect by serving a group of users of decaffeinated coffee two cups of regular coffee every morning in their office for a month, but have them drink decaf the rest of the time. The vigilance test would be given shortly after the coffee, and tolerance would be seen by an increase in errors over days. At the end of the month, they would be tested after drinking caffeinated coffee in the same and in a different setting.
The important points here are:
- Tolerance is shown by an increase in errors on the vigilance task.
- To see the effect of context, subjects need to be presented with caffeine in two different contexts.
- There needs to be a difference between the vigilance performance in the two contexts.
1.3 Contexts affects people’s response to alcohol, to off-color jokes, or to observed aggressive behavior.
1.5 The sample would be the addicts that we observe.
1.7 Not all people in the city are listed in the phone book. In particular, women and children are underrepresented. A phone book is particularly out of date as a random selection device with the increase in the use of cell phones.
Many telephone surveys really miss the general population, and instead focus on a restricted population, dominated by male adults.
1.9 In the tolerance study discussed in the text, we really do not care what the mean length of paw-lick latency is. No one would be excited to know that a mouse can stand on a surface at 105 degrees for 3.2 seconds without licking its paws. But we do very much care that the population mean of paw-lick latencies for morphine-tolerant mice is longer in one context than in another.
1.11 I would expect that your mother would continue to wander around in a daze, wondering what happened.
1.13 Three examples of measurement data: performance on a vigilance task; typing speed, blood alcohol level.
1.15 Relationship: The relationship between stress and susceptibility to disease; the relationship between driving speed and accident rate.
1.17 You could have one group of mice trained and tested in the same condition, one group trained in one condition and tested in the other, and a group given a placebo in the training context but given morphine in the testing condition.
1.19 This is an Internet search exercise without a fixed answer. The Statistics Homepage is an online statistics text. Various departments offer data sets, computing advice, and clarifying examples.
Chapter 2-Basic Concepts
2.1 Nominal: names of students in the class; Ordinal: the order in which students hand in their first exam; Interval: the student’s grade on that first exam; Ratio: the amount of time that the student spent studying for that exam.
2.3 If the rat lies down to sleep in the maze, after performing successfully for several trials, this probably says little about what the animal has learned in the task. It may say more about the animals level of motivation.
In this exercise I am trying to get the students to see that there is often quite a difference between what you and I think our variable is measuring and what it actually measures. Just because we label something as a measure of learning does not make it so. Just because the numbers increase on a ratio scale (twice as much time in the maze) doesn’t mean that what those numbers are actually measuring is ratio (twice as much learning).
2.5 We have to assume the following at the very least (and I am sure I left out some)
- Mice are adequate models for human behavior.
- Morphine tolerance effects in mice are like heroin tolerance effects in humans,
- Time on a warm surface is in some way analogous to a human response to heroin.
- A context shift for mice is analogous to a context shift for humans.
- A drug overdose is analogous to pain tolerance.
2.7 The independent variables are the sex of the subject and the sex of the other person.
2.9 The experimenter expected to find that women would eat less in the presence of a male partner than in the presence of a female partner. Men, on the other hand, were not expected to vary the amount that they ate as a function of sex of their partner.
2.11 We would treat a discrete variable as if it were continuous if it had many different levels and were at least ordinal.
2.13 When I drew 50 numbers 3 times I obtained 29, 26, and 19 even numbers, respectively. For my third drawing only 38 percent of my numbers were even, which is probably less than I might have expected—especially if I didn’t have a fair amount of experience with similar exercises.
2.15 Eyes level condition:
a) X3 = 2.03; X5 = 1.05; X8 = 1.86
b) ∑X = 14.82
c)
2.17 Eyes level condition:
a) (∑X)2 = 14.822 = 219.6324; ∑X2 = 1.652 + ... + 1.732 = 23.22
b) ∑X/N = 14.82/10 = 1.482
c) This is the mean, a type of average.
The above answers are the variance and standard deviation of Y. You really aren’t going to do much more calculation that this.
2.19 Putting the two sets of data together:
a) Multiply pairwise
b) ∑XY = 22.27496
c) ∑X∑Y = 14.82*14.63 = 216.82
d) SXY ≠ SXSY. They do differ, as you would expect.
e)
2.21 X 5 7 3 6 3 ∑X = 24
X + 4 9 11 7 10 7 ∑(X + 4) = 44 = (24 + 5*4)
2.23 In the text I spoke about room temperature as an ordinal scale of comfort (at least up to some point). Room temperature is a continuous measure, even though with respect to comfort it only measures at an ordinal level.
2.25 The Beth Perez story:
a) The dependent variable is the weekly allowance, measured in dollars and cents, and the independent variable is the sex of the child.
b) We are dealing with a selected sample—the children in her class.
c) The age of the students would influence the overall mean. The fact that these children are classmates could easily lead to socially appropriate responses—or what the children deem to be socially appropriate in their setting.
d) At least within her school, Beth could randomly sample by taking a student roster, assigning each student a number, and matching those up with numbers drawn from a random number table. Random assignment to Sex would obviously be impossible.
e) I don’t see negative aspects of the lack of random assignment here because that is the nature of the variable under consideration. It would be better if we could randomly assign a child to a sex and see the result, but we clearly can’t.
f) The outcome of the study could be influenced by the desire of some children to exaggerate their allowance, or to minimize it so as not to appear too different from their peers. I would suspect that boys would be likely to exaggerate.
g) The descriptive features of the study are her statements that the boys in her class received $3.18 per week in allowance, on average, while the girls received an average of $2.63. The inferential aspects are the inferences to the population of all children, concluding that “boys” get more than “girls.”
2.27 I would record the sequence number of each song that is played and then plot them on a graph. I can’t tell if they are truly random, but if I see a pattern to the points I can be quite sure that they are not random.
I think that it is important for students to become involved with the Internet early on. There is so much material out there that will be helpful, and you have to start finding it now. I find it impossible to believe that my explanations of concepts are always the best explanations that could be given and that they serve each student equally well. If one explanation doesn’t make sense, you can find others that may.
Chapter 3-Displaying Data
3.1 Katz et al (1990) No Passage Group:
There is too little data to say very much about the shape of this distribution, but it certainly isn’t looking normally distributed.
3.3 I would use stems of 3*, 3., 4*, 4. 5*, and 5. for this display.
3.5 Compared to those who read the passages:
a) Almost everyone who read the passages did better than the best person who did not read them. Certainly knowing what you are talking about is a good thing (though not always practiced).
4 | 3* |
b) 68966 | 3. |
44343 | 4* |
6669697 | 4. |
42102 | 5* |
57557 | 5. | 5669
| 6* |
| 6. | 66
| 7* | 21232231
| 7. | 5
| HI | 91 93
Notice that I have entered the data in the order in which I encountered
them, rather than in increasing order. It makes it easier.
c) It is obvious that the two groups are very different in their performance. We would be worried if they weren’t.
d) This is an Internet exercise with no fixed answer. That source is far more advanced than the students would be at this time, but I think that they should be able to read it if they just skip over what they don’t understand.
3.7 The following is a plot (as a histogram) of reaction times collapsed across all variables.
3.9 Histogram of GPA scores
3.11 (1) Mexico has very many young people and very few old people, while Spain has a more even distribution. (2) The difference between males and females is more pronounced at most ages in Spain than it is in Mexico. (3) You can see the high infant mortality rate in Mexico.
3.13 The distribution of those whose attendance is poor is far more spread out than the distribution of normal attendees. This would be expected because a few very good students can score well on tests even when they don’t attend, but most of the poor attenders are generally poor students who would score badly no matter what. The difference between the average grades of these two groups is obvious.