Statistics and Data Analysis For Behavioral Sciences

Dana S Dunn

Chapter 10 MEAN COMPARISON I: THE t TEST - all with Video Answers

Educators

Chapter Questions

00:51

Problem 1

Why is the $t$ test used in place of the $z$ test?

Sheryl Ezze

Numerade Educator

02:02

Problem 2

How do $t$ distributions differ from the $z$ distribution?

Abdullah Alomair

Numerade Educator

00:51

Problem 3

List the assumptions underlying use of a $t$ test. What happens when one of these assumptions is violated? Can the $t$ test still be used-why or why not?

Hossam Mohamed

Numerade Educator

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What does it mean when a statistical test is described as "robust"? Is the $t$ test robust? Under what particular conditions does use of the $t$ test become problematic, that is, a result based on it becomes less robust?

Shu Naito

Numerade Educator

01:54

Problem 5

Why are larger samples desirable? How do larger samples influence the size of a sample's standard deviation and standard error?

Yingtai Xiao

Numerade Educator

01:20

Problem 6

When should a researcher use the single sample $t$ test? Describe a hypothetical but concrete example.

Lucas Finney

Numerade Educator

01:38

Problem 7

A personality researcher wants to know if students attending smaller colleges are more introverted that those going to larger universities. The researcher selects a sample of $N=32$ 19 -year-old students from a small liberal arts college. These students complete a standardized introversion-extroversion scale, one whose scale characteristics were developed using university populations (lower scores indicate higher levels of introversion). The scale's $\mu=65$. The mean of the sample $\bar{X}=62$ and the standard deviation is 7 . Using a significance level of .05 , can the personality researcher assume that the college students are more introverted than the general population?

Dominador Tan

Numerade Educator

01:32

Problem 8

What is $95 \%$ confidence interval for the mean reported in problem 7.

Monique Whittaker

Numerade Educator

01:55

Problem 9

An organization of middle school educators believes that students' geographical knowledge has improved over the past five years. To verify this belief, a sample of 26 middle school students completes a world geography test $(\bar{X}=53, s=8.5)$. The middle school average on this measure is 50 . Using a significance level of .01 , have the educators demonstrated that geographical knowledge is actually improving? Why or why not?

Christopher Stanley

Numerade Educator

01:32

Problem 10

What is the $99 \%$ confidence interval for the mean reported in problem 9.

Monique Whittaker

Numerade Educator

04:24

Problem 11

A random sample is drawn and observed to have a $\bar{X}=45$ and an $s=11$. Use this sample data to test the null hypothesis that $\mu=42$ when:
a. The sample size is 40 and the significance level for the test is .05 .
b. The sample size is 25 and the significance level for the test is .01 .
c. The sample size is 62 and the significance level is .05 .

Lucas Finney

Numerade Educator

02:16

Problem 12

A random sample is drawn and observed to have an $X=77$ and an $s=7.5$. Use this sample data to test the null hypothesis that $\mu=80$ when:
a. The sample size is 30 and the significance level for the test is 01 .
b. The sample size is 43 and the significance level for the test is .05.
c. The sample size is 20 and the significance level is 05 .

Nick Johnson

Numerade Educator

03:21

Problem 13

A campus psychologist is starting a weekly group to discuss low level depression, the type often triggered by stressful, academic events. The psychologist knows that the population mean of the screening test is 35 , and that scores at or above this level indicated the probable presence of low level depression. His sample of students obtained screening test scores of $30,36,34,34,38,40,32,30,28,37,36$, and 37. Can the psychologist label this group of students as having low level depression so that he can begin the therapy group? Use a .05 level of significance and perform a one-tailed test.

Sheryl Ezze

Numerade Educator

00:24

Problem 14

A teacher of gifted students believes that her sample of students have IQs still higher than the superior gifted cutoff of 134. Her students' scores on the IQ test were 132, 130, $136,137,135,136,133,135$, and 137 . Is the teacher correct? Can she claim that as a group, her students exceed the superior level? Use a significance level of .01 and perform a one-tailed test.

Hossam Mohamed

Numerade Educator

01:42

Problem 15

What are the confidence intervals for the sample means based on the data reported in problems 13 and 14. Base the confidence interval on the appropriate significance level provided in each problem.

Neel Faucher

Numerade Educator

Problem 16

Discuss the factors affecting the statistical power of a $t$ test.

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Problem 17

Why is the $t$ test for independent groups ideal for hypothesis testing in experimental research?

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00:14

Problem 18

How do traditional independent variables differ from subject variables?

Becky Rahm

Numerade Educator

Problem 19

What is a subject variable? How are subject variables used in concert with between-groups designs and the independent groups $t$ test?

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00:58

Problem 20

How does the standard error of the difference between means differ from the usual measures of standard error?

Carly Stoner

Numerade Educator

Problem 21

Explain the nature of the conceptual model for comparing means presented in this chapter. Why is this model an appropriate prelude for most inferential statistical tests?

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02:07

Problem 22

Unbeknownst to his class, an instructor decides to replicate a classic study on experimenter expectancy effects (Rosenthal & Fode, 1963). In an experimental psychology lab, each student was given a rat to teach to run a standard maze. However, half of the students were told their rats were specially bred to be "maze bright" while the remaining students were told their rats were "maze dull." In actuality, of course, all the rats were of the same breed and possessed no special talents. The following data represent the rats' skill-level at running the mazes, where lower numbers represent fewer errors (i.e., higher skill). Did the students with the maze bright rats transmit that expectancy to the animals, so that they outperformed the "maze dull" animals? Use a one-tailed test with a significance level of .05 .
Maze bright scores: $15,10,11,10,12,13,10,13,12,11$
Maze dull scores: $17,18,17,16.5,17,19,13,12,18,17$

Sheryl Ezze

Numerade Educator

02:42

Problem 23

What is the effect size for the result obtained in problem 22 ? If the $t$ statistic calculated in problem 22 reached significance, then determine the value of $\hat{\omega}^2$.

Ameer Said

Numerade Educator

01:17

Problem 24

Using the examples presented earlier in the chapter, write a paragraph or two summarizing the results obtained in problems 22 and 23.

Lucas Finney

Numerade Educator

02:02

Problem 25

Participants took part in a study on aversive noise and its affect on performing skilled tasks. An experimental psychologist believes that the predictability of the aversive noise is key to understanding its influence on performance. Specifically, predictable noise-noise occurring at fixed intervals of time-is less disruptive than random noise, which reminds participants that they lack control in the situation. Two groups of participants solved moderately difficult math problems while listening to a loud noise. One group heard the loud noise at fixed intervals, the other heard it at random intervals. The following data are the total number of math problems that participants in each group got correct. Determine whether the experimenter was right, that random noise is more distuptive to skilled performance than fixed noise (use a .05 significance level for a one-tailed test).
Random noise: $10,9,8,8,9,10,9,8,7,8$.
Fixed noise: $6,7,8,8,7,7,6,5,8,6$.

Sheryl Ezze

Numerade Educator

Problem 26

What is the effect size for the result obtained in problem 25 ? If the $t$ statistic calculated in problem 25 reached significance, then determine the value of $\hat{\omega}^2$.

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00:41

Problem 27

Using the examples presented earlier in the chapter, write a paragraph or two summarizing the results obtained in problems 25 and 26.

Wendi Zhao

Numerade Educator

06:44

Problem 28

When teaching about science, instructors are assumed to rely on less descriptive or flowery language because the natural sciences contain topical information that is relatively fixed in content. Classroom instructors in the humanities, however, rely on much more descriptive language because the topics are open to debate and multiple interpretations. An educational rescarcher sits in several natural science and humanities classes at several high schools and monitors the teaching styles of instructors. What follows are the number of different words and phrases used by the respective sets of instructors to convey course material: Did the humanities teachers use more descriptive language than the natural science instructors? Use a .01 significance level and perform the appropriate one-tailed test, and take note that the groups are not of equal size.
Humanities instructors' speech: $34,28,27,32,33,30,38,32$, $30,28,34,32$
Natural science teachers' speech: 27, 26, 23, 30, 25, 26, 27, 26, 27

Aimal Hassan

Numerade Educator

Problem 29

What is the effect size for the result obtained in problem 28 ? If the $t$ statistic calculated in problem 28 reached significance, then determine the value of $\hat{\omega}^2$.

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01:09

Problem 30

Using the examples presented earlier in the chapter, write a paragraph or two summarizing the results obtained in problems 28 and 29.

Carson Merrill

Numerade Educator

01:14

Problem 31

Are there any advantages to conducting a correlated groups design rather than an independent groups design? If so, what are they?

Jameson Kuper

Numerade Educator

Problem 32

How does a correlated groups design differ from a matched groups design? When is it appropriate to use one design rather than the other?

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Problem 33

What is a carryover effect? Why do such effects pose concerns for correlated groups designs?

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07:58

Problem 34

An educational psychologist believes that relaxation training can help students to improve their performance on standardized tests. A group of high school students completes two grade-appropriate reading comprehension tests, one before the relaxation training and the other a week later, after the training is complete. Using the following data, demonstrate whether reading comprehension scores increased following the training. Use a significance level of .05 as well as a one-tailed test.
$$
\begin{array}{lcc}
\text { Participant } & \text { Test Score }_1 & \text { Test Score }_2 \\
\hline \text { A } & 8 & 10 \\
\text { B } & 6 & 7 \\
\text { C } & 6 & 8 \\
\text { D } & 5 & 6 \\
\text { E } & 9 & 10 \\
\text { F } & 8 & 9 \\
\hline
\end{array}
$$

James Kiss

Numerade Educator

00:31

Problem 35

Review the research project described above in problem 34 . Do you think this project could be susceptible to any carryover effects? If so, which one(s) and why?

R M

Numerade Educator

02:35

Problem 36

An industrial psychologist is concerned that a recent round of layoffs at a plant may have increased the stress felt by employees who retained their positions. To measure whether this survivor stress actually exists, the psychologist administered a stress measure to a sample of employees before and after the layoffs occurred. Evaluate whether self-reported stress increased from time $e_1$ to time $e_2$ by using a .01 significance level and a one-tailed test.
$$
\begin{array}{lcc}
\text { Employee } & \text { Stress Score }_1 & \text { Stress }_1 \text { Score }_2 \\
\hline \text { A } & 21 & 23 \\
\text { B } & 24 & 26 \\
\text { C } & 15 & 15 \\
\text { D } & 18 & 22 \\
\text { E } & 19 & 20 \\
\text { F } & 20 & 19 \\
\text { G } & 18 & 21 \\
\hline
\end{array}
$$

Sheryl Ezze

Numerade Educator

00:31

Problem 37

Review the research project described above in problem 36. Do you think this project could be susceptible to any carryover effects? If so, which one(s) and why?

R M

Numerade Educator

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Problem 38

A social psychologist is interested in how emotional contagion-adopting the affective state of people after hearing them describe positive or negative experience-affects people who have an optimistic disposition. The psychologist is particularly interested in whether the number of people sharing their emotional experiences makes any difference in the affective transfer (e.g.n perhaps two people sharing a happy event with a third enhance her mood to a greater degree than one person talking about the same experience). The social psychologist recruits a group of people who are matched on their measured level of optimism, age, and gender. One member of each pair is then exposed to three people who discuss the same emotion eliciting experience, while the other matched participant meets with one person for emotion sharing. The following data are emotion ratings on a 1 to 7 scale, where higher numbers indicate greater emotion transfer. Evaluate whether the participants who met with a group rather than a single person showed a greater degree of emotional contagion. Use a .05 significance level for a onetailed hypothesis test.
$$
\begin{array}{lcc}
\text { Matched Pair } & \begin{array}{l}
\text { Solo-Emotion } \\
\text { Encounter }
\end{array} & \begin{array}{c}
\text { Group Emotional } \\
\text { Encounter }
\end{array} \\
\hline \text { A } & 4 & 6 \\
\text { B } & 3 & 5 \\
\text { C } & 7 & 6 \\
\text { D } & 5 & 7 \\
\text { E } & 2 & 4 \\
\text { F } & 6 & 6 \\
\text { G } & 5 & 7 \\
\text { H } & 5 & 6 \\
\hline
\end{array}
$$

Rashmi Sinha

Numerade Educator

01:39

Problem 39

Why should investigators learn to perform a power analysis? Should a power analysis be performed before or after a study? Why?

Shazia Naz

Numerade Educator

Problem 40

Examine the following effect sizes and power levels, and then determine how many total participants are needed for each of the studies represented.
a. effect $r=.30$, power $=.40$; b. effect size $r=.20$, power $=.60 ;$ c. effect size $r=.70$, power $=.15$; d. effect size $r=.30$, power $=.50$.

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01:12

Problem 41

Which $t$ test is most appropriate for each of the following situations?
a. Samples are not independent of one another.
b. Population parameters are known.
c. Two observations at different points in time were gathered for each participant.
d. Two independent samples were drawn.
e. One observation was drawn for each participant, and population parameters were not known.

Jake Zanazzi

Numerade Educator

00:52

Problem 42

Two independent samples of data are available, but their sample sizes are unequal. What should the data analyst do?

Maxime Rossetti

Numerade Educator