Statistics Learning From Data

Thomas H. Short, Roxy Peck

Chapter 16 Asking and Answering Questions About More Than Two Means - all with Video Answers

Educators

Section 1

The Analysis of Variance—Single-Factor ANOVA and the F Test

11:30

Problem 1

Give as much information as you can about the $P$ -value for an upper-tailed $F$ test in each of the following situations.
a. $\mathrm{df}_{1}=4, \mathrm{df}_{2}=15, F=5.37$
b. $\mathrm{df}_{1}=4, \mathrm{df}_{2}=15, F=1.90$
c. $\mathrm{df}_{1}=4, \mathrm{df}_{2}=15, F=4.89$
d. $\mathrm{df}_{1}=3, \mathrm{df}_{2}=20, F=14.48$
e. $\mathrm{df}_{1}=3, \mathrm{df}_{2}=20, F=2.69$
f. $\mathrm{df}_{1}=4, \mathrm{df}_{2}=50, F=3.24$

Sonam Khatri

Numerade Educator

04:53

Problem 2

Employees of a certain state university system can choose from among four different health plans. Each plan differs somewhat from the others in terms of hospitalization coverage. Four random samples of recently hospitalized individuals were selected, each sample consisting of people covered by a different health plan. The length of the hospital stay (number of days) was determined for each individual selected.
a. What hypotheses would you test to decide whether the mean lengths of stay are not the same for all four health plans?
b. If each sample consisted of eight individuals and the value of the ANOVA $F$ statistic was $F=4.37$, what conclusion would be appropriate for a test with $\alpha=0.01 ?$
c. Answer the question posed in Part (b) if the $F$ value given there resulted from sample sizes $n_{1}=9, n_{2}=8, n_{3}=7$, and $n_{4}=8$.

Jameson Kuper

Numerade Educator

01:43

Problem 3

The authors of the paper "Age and Violent Content Labels Make Video Games Forbidden Fruits for Youth” (Pediatrics [2009]: 870-876) carried out an experiment to determine if restrictive labels on video games actually increased the attractiveness of the game for young game players. Participants read a description of a new video game and were asked how much they wanted to play the game. The description also included an age rating. Some participants read the description with an age restrictive label of $7+$, indicating that the game was not appropriate for children under the age of 7 . Others read the same description, but with an age restrictive label of $12+, 16+,$ or $18+$. The following data for 12- to 13-year-old boys are consistent with summary statistics given in the paper. (The sample sizes in the actual experiment were larger.) For purposes of this exercise, you can assume that the boys were assigned at random to one of the four age label treatments $(7+, 12+, 16+,$ and $18+) .$ Data shown are the boys' ratings of how much they wanted to play the game on a scale of 1 to 10 . Do the data provide convincing evidence that the mean rating associated with the game description by 12 - to 13-year-old boys is not the same for all four restrictive rating labels? Test the appropriate hypotheses using a significance level of 0.05

Dominador Tan

Numerade Educator

11:55

Problem 4

The authors of the paper "Reading Subtitles and Taking Enotes While Learning Scientific Materials in a Multimedia Environment" (Educational Technology and Society [2016): 47-58) were interested in determining if including subtitles and providing opportunities to take electronic notes while listening to online materials would enhance learning for students whose first language was not English. Students were randomly assigned to one of four groups. In the first group, subtitles were included in the online materials that the students were asked to study, but the ability to take electronic notes was not provided. For the second group, no subtitles were provided but students were able to take electronic notes. For the third group, both subtitles and the ability to take electronic notes were available. Students in the fourth group did not have access to either subtitles or the ability to take electronic notes. After studying the online materials, all students took a 14 -question test on the material studied. Minitab output based on data consistent with summary quantities in the paper is shown on the next page. Is there evidence to conclude that the mean test score differs for at least two of the treatments? Use the given computer output to test the appropriate hypotheses with a significance level of 0.05

Mohan Jain

Numerade Educator

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Problem 5

Give as much information as you can about the $P$ -value of the single-factor ANOVA $F$ test in each of the following situations.
a. $k=5, n_{1}=n_{2}=n_{3}=n_{4}=n_{5}=4, F=5.37$
b. $k=5, n_{1}=n_{2}=n_{3}=5, n_{4}=n_{5}=4, F=2.83$
c. $k=3, n_{1}=4, n_{2}=5, n_{3}=6, F=5.02$
d. $k=3, n_{1}=n_{2}=4, n_{3}=6, F=15.90$
e. $k=4, n_{1}=n_{2}=15, n_{3}=12, n_{4}=10, F=1.75$

Victor Salazar

Numerade Educator

07:18

Problem 6

The paper referenced in Exercise 16.3 also gave data for 12 - to 13-year-old girls. Data consistent with summary values in the paper are shown below. Do the data provide convincing evidence that the mean rating associated with the game description for 12 - to 13 -year-old girls is not the same for all four age restrictive rating labels? Test the appropriate hypotheses using $\alpha=0.05$.

Erin Moser

Numerade Educator

01:48

Problem 7

The experiment described in Example 16.4 also gave data on change in body fat mass for men ("Growth Hormone and Sex Steroid Administration in Healthy Aged Women and Men," Journal of the American Medical Association [2002]: 2282-2292). Each of 74 male subjects who were over age 65 was assigned at random to one of the following four treatments: (1) placebo "growth hormone" and placebo "steroid" (denoted by $\mathrm{P}+\mathrm{P}$ ); (2) placebo "growth hormone" and the steroid testosterone (denoted by $\mathrm{P}+\mathrm{S}$ ); (3) growth hormone and placebo "steroid" (denoted by $\mathrm{G}+\mathrm{P}$ ); and (4) growth hormone and the steroid testosterone (denoted by $\mathrm{G}+\mathrm{S}$ ). The accompanying table gives data on change in body fat mass over the 26-week period following the treatment that are consistent with summary quantities given in the article.
Also, $N=74$, grand total $=-158.3$, and the mean of all 74 observations is $\overline{\bar{x}}=\frac{-158.3}{74}=-2.139$ Carry out an $F$ test to see whether mean change in body fat mass differs for the four treatments.

James Kiss

Numerade Educator

03:10

Problem 8

In an experiment to investigate the performance of four different brands of spark plugs intended for use on a $125-\mathrm{cc}$ motorcycle, five plugs of each brand were tested, and the number of miles (at a constant speed) until failure was observed. A partially completed ANOVA table is given. Fill in the missing entries, and test the relevant hypotheses using a 0.05 level of significance.

Shu Naito

Numerade Educator

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Problem 9

Do people feel hungrier after sampling a healthy food? The authors of the paper "When Healthy Food Makes You Hungry" (Journal of Consumer Research [2010]: S34-S44) carried out a study to answer this question. They randomly assigned volunteers into one of three groups. The people in the first group were asked to taste a snack that was billed as a new health bar containing high levels of protein, vitamins, and fiber. The people in the second group were asked to taste the same snack but were told it was a tasty chocolate bar with a raspberry center. After tasting the snack, participants were asked to rate their hunger level on a scale from 1 (not at all hungry) to 7 (very hungry). The people in the third group were asked to rate their hunger but were not given a snack. The data in the accompanying table are consistent with summary quantities given in the paper (although the sample sizes in the actual study were larger).
a. Do these data provide evidence that the mean hunger rating differs for at least two of the treatments ("healthy" snack, "tasty" snack, no snack)? Test the relevant hypotheses using a significance level of 0.05 .
b. Is it reasonable to conclude that the mean hunger rating is greater for people who do not get a snack? Explain.

Rashmi Sinha

Numerade Educator

02:37

Problem 10

The accompanying summary statistics for a measure of social marginality for samples of youths, young adults, adults, and seniors appeared in the paper "Perceived Causes of Loneliness in Adulthood" (Journal of Social Behavior and Personality [2000]: 67-84). The social marginality score measured actual and perceived social rejection, with higher scores indicating greater social rejection. For purposes of this exercise, assume that it is reasonable to regard the four samples as representative of the U.S. population in the corresponding age groups and that the distributions of social marginality scores for these four groups are approximately normal with the same standard deviation. Is there evidence that the mean social marginality score differs for at least two of the four age groups? Test the relevant hypotheses using $\alpha=0.05$.

Sheryl Ezze

Numerade Educator

03:12

Problem 11

The chapter Preview Example described a study comparing three groups of college students (soccer athletes, nonsoccer athletes, and a comparison group consisting of students who did not participate in intercollegiate sports). The following is information on scores from the Hopkins Verbal Learning Test (which measures immediate memory recall). In addition, $\overline{\bar{x}}=30.19$ Suppose that it is reasonable to regard these three samples as random samples from the three student populations of interest. Is there sufficient evidence to conclude that the mean Hopkins score is not the same for the three student populations? Use $\alpha=0.05$.

Sheryl Ezze

Numerade Educator

01:34

Problem 12

It is common for baseball pitchers to use stretching to prepare for a game. But does this make a difference? The authors of the paper "The Acute Effects of Upper Extremity Stretching on Throwing Velocity in Baseball Throwers" (Journal of Sports Medicine [2013]: 1-7) carried out an experiment to compare two different types of stretching and a control treatment consisting of no stretching. Participants were adult males with varying levels of baseball throwing experience and who were not professional or collegiate baseball players. Participants in the two stretching treatments went through a warm-up that included 8 minutes of stretching. Each participant (all three groups) then threw 10 pitches, and the average speed (km/hour) was calculated.
a. Explain why it is important that the participants be assigned at random to the three different treatment groups (Stretching Method 1 , Stretching Method 2 , and No Stretching).
b. The following computer output and summary values are based on simulated data that are consistent with information and conclusions given in the paper. Use the given output to determine if there is evidence to support the claim that the mean average speed is not the same for all three treatments. Use a significance level of 0.05 for your test.
c. Previous research on the effect of stretching on performance in other sports, such as running, has concluded that stretching can improve performance. Why do you think that the authors of this paper were surprised by the results of this study?

Dominador Tan

Numerade Educator

11:59

Problem 13

Parents are frequently concerned when their child seems slow to begin walking (although when the child finally walks, the resulting havoc sometimes has the parents wishing they could turn back the clock!). The article "Walking in the Newborn" (Science, 176 [1972]: $314-315$ ) reported on an experiment in which the effects of several different treatments on the age at which a child first walks were compared. Children in the first group were given special walking exercises for 12 minutes per day beginning at age 1 week and lasting 7 weeks. The second group of children received daily exercises but not the walking exercises administered to the first group. The third and fourth groups were control groups. They received no special treatment and differed only in that the third group's progress was checked weekly, whereas the fourth group's progress was checked just once at the end of the study. Observations on age (in months) when the children first walked are shown in the accompanying table. Also given is the ANOVA table, obtained from the SPSS computer package.
a. Verify the entries in the ANOVA table.
b. State and test the relevant hypotheses using a significance level of 0.05

James Kiss

Numerade Educator

03:55

Problem 14

Leaf surface area is an important variable in plant gas-exchange rates. Dry matter per unit surface area (mg/ $\mathrm{cm}^{3}$ ) was measured for trees raised under three different growing conditions. Let $\mu_{1}, \mu_{2},$ and $\mu_{3}$ represent the mean dry matter per unit surface area for the growing conditions $1,2,$ and $3,$ respectively. Suppose that the given $95 \% \mathrm{~T}-\mathrm{K}$ confidence intervals are:
$\begin{array}{lccc}\text { Difference } & \mu_{1}-\mu_{2} & \mu_{1}-\mu_{3} & \mu_{2}-\mu_{3} \\ \text { Interval } & (-3.11,-1.11) & (-4.06,-2.06) & (-1.95,0.05)\end{array}$
Which of the following four statements do you think describes the relationship between $\mu_{1}, \mu_{2},$ and $\mu_{3} ?$ Explain your choice.
a. $\mu_{1}=\mu_{2},$ and $\mu_{3}$ differs from $\mu_{1}$ and $\mu_{2}$.
b. $\mu_{1}=\mu_{3},$ and $\mu_{2}$ differs from $\mu_{1}$ and $\mu_{3}$
c. $\mu_{2}=\mu_{3},$ and $\mu_{1}$ differs from $\mu_{2}$ and $\mu_{3}$
d. All three $\mu$ 's are different from one another.

Benjamin Chaback

Numerade Educator

01:39

Problem 15

The accompanying underscoring pattern appears in the article "Women's and Men's Eating Behavior Following Exposure to Ideal-Body Images and Text"(Communications Research [2006]: 507-529). Women either viewed slides depicting images of thin female models with no text (treatment 1); viewed the same slides accompanied by diet and exercise-related text (treatment 2); or viewed the same slides accompanied by text that was unrelated to diet and exercise (treatment 3). A fourth group of women did not view any slides (treatment 4). Participants were assigned at random to the four treatments. Participants were then asked to complete a questionnaire in a room where pretzels were set out on the tables. An observer
recorded how many pretzels participants ate while completing the questionnaire. Write a few sentences interpreting this underscoring pattern.
$\begin{array}{lcccc}\text { Treatment: } & 2 & 1 & 4 & 3 \\ \text { Mean number of pretzels } & 0.97 & 1.03 & 2.20 & 2.65 \\ \text { consumed: } & & & & \end{array}$

James Kiss

Numerade Educator

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Problem 16

The following data resulted from a flammability study in which specimens of five different fabrics were tested to determine burn times (in seconds).
$\begin{aligned} \mathrm{MSTr} &=23.67 \\ \mathrm{MSE} &=1.39 \\ F &=17.08 \\ P \text { -value } &=0.000 \end{aligned}$
The accompanying output gives the T-K intervals as calculated by Minitab. Identify significant differences and give the underscoring pattern.

Shu Naito

Numerade Educator

03:35

Problem 17

The paper "Trends in Blood Lead Levels and Blood Lead Testing among U.S. Children Aged 1 to 5 Years" (Pediatrics [2009]: e376-e385) gave data on blood lead levels (in $\mathrm{mg} / \mathrm{dL}$ ) for samples of children living in homes that had been classified either at low, medium, or high risk of lead exposure, based on when the home was constructed. After using a multiple comparison procedure, the authors reported the following:
1. The difference in mean blood lead level between low-risk housing and medium-risk housing was significant.
2. The difference in mean blood lead level between low-risk
housing and high-risk housing was significant.
3. The difference in mean blood lead level between mediumrisk housing and high-risk housing was significant.
Which of the following sets of T-K intervals (Set $1,2,$ or 3$)$ is consistent with the authors' conclusions? Explain your choice.
$\mu_{L}=$ mean blood lead level for children living in low-risk housing $\mu_{M}=$ mean blood lead level for children living in mediumrisk housing $\mu_{H}=$ mean blood lead level for children living in high-risk housing

Hossam Mohamed

Numerade Educator

01:39

Problem 18

The paper referenced in the Exercise 16.15 also gave the following underscoring pattern for men.
a. Write a few sentences interpreting this underscoring pattern.
b. Using your answers from Part (a) and from the Exercise $16.15,$ write a few sentences describing the differences between how men and women respond to the treatments.

James Kiss

Numerade Educator

04:05

Problem 19

Do lizards play a role in spreading plant seeds? Some research carried out in South Africa would suggest so ("Dispersal of Namaqua Fig [Ficus cordata cordata] Seeds by the Augrabies Flat Lizard [Platysaurus broadleyi]," Journal of Herpetology [1999]: 328-330). The researchers collected 400 seeds of a particular type of fig, 100 of which were from each treatment: lizard dung, bird dung, rock hyrax dung, and uneaten figs. They planted these seeds in batches of $5,$ and for each group of 5 they recorded how many of the seeds germinated. This resulted in 20 observations for each treatment. The treatment means and standard deviations are given in the accompanying table.
a. Construct the appropriate ANOVA table, and test the hypothesis that there is no difference between mean number of seeds germinating for the four treatments.
b. Is there evidence that seeds eaten and then excreted by lizards germinate at a higher rate than those eaten and then excreted by birds? Give statistical evidence to support your answer.

Jon Southam

Numerade Educator

08:36

Problem 20

Suppose that samples of six different brands of diet or imitation margarine were analyzed to determine the level of physiologically active polyunsaturated fatty acids (PAPUFA, in percent), resulting in the accompanying data.
$$
\begin{array}{llllll}
\text { Imperial } & 14.1 & 13.6 & 14.4 & 14.3 & \\
\text { Parkay } & 12.8 & 12.5 & 13.4 & 13.0 & 12.3 \\
\text { Blue Bonnet } & 13.5 & 13.4 & 14.1 & 14.3 & \\
\text { Chiffon } & 13.2 & 12.7 & 12.6 & 13.9 & \\
\text { Mazola } & 16.8 & 17.2 & 16.4 & 17.3 & 18.0 \\
\text { Fleischmann's } & 18.1 & 17.2 & 18.7 & 18.4 &
\end{array}
$$
a. Carry out a test to determine if there is evidence of differences among the true mean PAPUFA percentages for the different brands. Use $\alpha=0.05$.
b. Use the T-K procedure to calculate $95 \%$ simultaneous confidence intervals for all differences between pairs of means and give the corresponding underscoring pattern.

Robin Corrigan

Numerade Educator

01:44

Problem 21

In an experiment to investigate the effect of the portrayal of female characters in superhero movies, researchers randomly assigned female college students to one of three groups ("The Empowering (Super) Heroine? The Effects of Sexualized Female Characters in Superhero Films on Women," Sex Roles [2015]: 211-220). One group was a control group, one group watched 13 minutes of video scenes from the movie Spider-Man (where a sexy female character was portrayed as a victim), and one group watched 13 minutes of video scenes from the movie $X$ -Men (where a sexy female character was portrayed as a heroine). The women in the control group did not watch a video. The women in all three groups then completed a questionnaire and their answers were used to calculate a measure of gender stereotyping, with lower values indicating attitudes more accepting of equality of women and men. The researchers used a one-way ANOVA to analyze the data. The following Minitab ANOVA output and summary statistics are based on data consistent with information and conclusions from the paper.
a. Use the given output to test the null hypothesis of no difference in mean gender stereotyping score for the three different treatment groups. Use a significance level of $0.05 .$
b. Minitab reported the following T-K intervals:
$$
\begin{array}{ll}
\text { Control }-\text { X-Men: } & (-0.819,0.114) \\
\text { Control - Spider Man: } & (-0.887,-0.017) \\
\text { X-Men - Spider Man : } & (-0.563,0.363)
\end{array}
$$
Use this information to construct the corresponding underscore pattern.
c. Write a few sentences describing what you learned from the results of the Tukey-Kramer procedure.

Dominador Tan

Numerade Educator

03:22

Problem 22

Consider the accompanying data on plant growth after the application of five different types of growth hormone.
$$
\begin{array}{lllll}
\mathbf{1} & 13 & 17 & 7 & 14 \\
\mathbf{2} & 21 & 13 & 20 & 17 \\
\mathbf{3} & 18 & 14 & 17 & 21 \\
\mathbf{4} & 7 & 11 & 18 & 10 \\
\mathbf{5} & 6 & 11 & 15 & 8
\end{array}
$$
a. Carry out the ANOVA $F$ test using a significance level of $\alpha=0.05$
b. What happens when the T-K procedure is applied? (Note:
This "contradiction" can occur when $H_{0}$ is "barely" rejected. It happens because the test and the multiple comparison method are based on different distributions. Consult your friendly neighborhood statistician for more information.)

Dominador Tan

Numerade Educator

10:18

Problem 23

The paper "Women's and Men's Eating Behavior Following Exposure to Ideal-Body Images and Text" (Communication Research [2006]: 507-529) describes an experiment in which 74 men were assigned at random to one of four treatments:
1. Viewed slides of fit, muscular men
2. Viewed slides of fit, muscular men accompanied by diet and fitness-related text
3. Viewed slides of fit, muscular men accompanied by text not related to diet and fitness
4. Did not view any slides
The participants then went to a room to complete a questionnaire. In this room, bowls of pretzels were set out on the tables. A research assistant noted how many pretzels were consumed by each participant while completing the questionnaire. Data consistent with summary quantities given in the paper are given in the accompanying table. Do these data provide convincing evidence that the mean number of pretzels consumed is not the same for all four treatments? Test the relevant hypotheses using a significance level of 0.05 .

Beth Stone

Numerade Educator

01:55

Problem 24

Can use of an online plagiarism-detection system reduce plagiarism in student research papers? The paper "Plagiarism and Technology: A Tool for Coping with Plagiarism" (Journal of Education for Business [2005]: 149-152) describes a study in which randomly selected research papers submitted by students during five semesters were analyzed for plagiarism. For each paper, the percentage of plagiarized words in the paper was determined by an online analysis. In each of the five semesters, students were told during the first two class meetings that they would have to submit an electronic version of their research papers and that the papers would be reviewed for plagiarism. Suppose that the number of papers sampled in each of the five semesters and the means and standard deviations for percentage of plagiarized words are as given in the accompanying table. For purposes of this exercise, assume that the conditions necessary for the ANOVA $F$ test are reasonable. Do these data provide evidence to support the claim that mean percentage of plagiarized words is not the same for all five semesters? Test the appropriate hypotheses using $\alpha=0.05$.

Nick Johnson

Numerade Educator

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Problem 25

The paper referenced in Exercise 16.3 described an experiment to determine if restrictive age labeling on video games increased the attractiveness of the game for boys ages 12 to $13 .$ In that exercise, the null hypothesis was $H_{0}: \mu_{1}=\mu_{2}$ $=\mu_{3}=\mu_{4},$ where $\mu_{1}$ is the population mean attractiveness rating for the game with the $7+$ age label, and $\mu_{2}, \mu_{3},$ and $\mu_{4}$ are the population mean attractiveness scores for the $12+, 16+,$ and $18+$ age labels, respectively. The sample data are given in the accompanying table.
a. Calculate the $95 \%$ T-K intervals and then use the underscoring procedure described in this section to identify significant differences among the age labels.
b. Based on your answer to Part (a), write a few sentences commenting on the theory that the more restrictive the age label on a video game, the more attractive the game is to 12 - to 13 -year-old boys.

Rashmi Sinha

Numerade Educator

02:00

Problem 26

The authors of the paper "Beyond the Shooter Game:
Examining Presence and Hostile Outcomes among Male Game Players" (Communication Research [2006]: 448-466) studied how video game content might influence attitudes and behavior. Male students at a large midwestern university were assigned at random to play one of three action-oriented video games. Two of the games involved some violenceone was a shooting game and one was a fighting game. The third game was a nonviolent race car driving game. After playing a game for 20 minutes, participants answered a set of questions. The responses were used to determine values of three measures of aggression: (1) a measure of aggressive behavior; (2) a measure of aggressive thoughts; and (3) a measure of aggressive feelings. The authors hypothesized that the means for the three measures of aggression would be greatest for the fighting game and lowest for the driving game.
a. For the measure of aggressive behavior, the paper reports that the mean score for the fighting game was significantly higher than the mean scores for the shooting and driving game, but that the mean scores for the shooting and driving games were not significantly different. The three sample means were:
Use the underscoring procedure of this section to construct a display that shows any significant differences in mean aggressive behavior score among the three games.
b. For the measure of aggressive thoughts, the three sample means were:
The paper states that the mean score for the fighting game only significantly differed from the mean score for the driving game, and that the mean score for the shooting game did not significantly differ from either the fighting or driving games. Use the underscoring procedure of this section to construct a display that shows any significant differences in mean aggressive thoughts score among the three games.

Nick Johnson

Numerade Educator