Statistics Informed Decisions Using Data

Michael Sullivan III

Chapter 13 Comparing Three or More Means - all with Video Answers

Educators

Section 1

Comparing Three or More Means

00:42

Problem 1

The acronym $ANOVA$ stands for _____ _ _____.

Neel Faucher

Numerade Educator

01:58

Problem 2

True or False: To perform a one-way ANOVA, the populations do not need to be normally distributed.

Neel Faucher

Numerade Educator

02:05

Problem 3

True or False: To perform a one-way ANOVA, the populations must have the same variance.

Neel Faucher

Numerade Educator

02:12

Problem 4

The variability among the sample means is called _____- sample variability, and the variability of each sample is the _____- sample variability.

Neel Faucher

Numerade Educator

02:21

The variability within each treatment group, which is a weighted average of the sample variances from each treatment where the weights are based on the size of each sample, is called the mean square due to _____ and is denoted _____.

Neel Faucher

Numerade Educator

01:23

Problem 6

True or False: The $F$ -test statistic is $F_{0}=\frac{\mathrm{MST}}{\mathrm{MSE}}$.

Neel Faucher

Numerade Educator

02:11

Problem 7

Fill in the ANOVA table.
$$\begin{array}{lcccc}
\text { Source of } & \text { Sum of } & \text { Degrees of } & \text { Mean } & \text { F-Test } \\
\text { Variation } & \text { Squares } & \text { Freedom } & \text { Squares } & \text { Statistic } \\
\hline \text { Treatment } & 387 & 2 & & \\
\hline \text { Error } & 8042 & 27 & & \\
\hline \text { Total } & & & &
\end{array}$$

Neel Faucher

Numerade Educator

01:42

Problem 8

Fill in the ANOVA table.
$$\begin{array}{lcccc}
\text { Source of } & \text { Sum of } & \text { Degrees of } & \text { Mean } & \text { F-Test } \\
\text { Variation } & \text { Squares } & \text { Freedom } & \text { Squares } & \text { Statistic } \\
\hline \text { Treatment } & 2814 & 3 & & \\
\hline \text { Error } & 4915 & 36 & & \\
\hline \text { Total } & & & &
\end{array}$$

Neel Faucher

Numerade Educator

04:29

Problem 9

Determine the F-test statistic based on the given summary statistics. /Hint: $\bar{x}=\frac{\Sigma n_{i} \bar{x}_{i}}{\Sigma n_{i}} .1$
$$\begin{array}{cccc}
\text { Population } & \begin{array}{c}
\text { Sample } \\
\text { Size }
\end{array} & \begin{array}{c}
\text { Sample } \\
\text { Mean }
\end{array} & \begin{array}{c}
\text { Sample } \\
\text { Variance }
\end{array} \\
\hline 1 & 10 & 40 & 48 \\
\hline 2 & 10 & 42 & 31 \\
\hline 3 & 10 & 44 & 25
\end{array}$$

Neel Faucher

Numerade Educator

04:22

Problem 10

Determine the F-test statistic based on the given summary statistics. /Hint: $\bar{x}=\frac{\Sigma n_{i} \bar{x}_{i}}{\Sigma n_{i}} .1$
$$\begin{array}{cccc}
\text { Population } & \begin{array}{c}
\text { Sample } \\
\text { Size }
\end{array} & \begin{array}{c}
\text { Sample } \\
\text { Mean }
\end{array} & \begin{array}{c}
\text { Sample } \\
\text { Variance }
\end{array} \\
\hline 1 & 15 & 105 & 34 \\
\hline 2 & 15 & 110 & 40 \\
\hline 3 & 15 & 108 & 30 \\
\hline 4 & 15 & 90 & 38
\end{array}$$

Neel Faucher

Numerade Educator

01:13

Problem 11

The following data represent a simple random sample of $n=4$ from three populations that are known to be normally distributed. Verify that the $F$ -test statistic is 2.04
$$\begin{array}{ccc}
\text { Sample 1 } & \text { Sample 2 } & \text { Sample 3 } \\
\hline 28 & 22 & 25 \\
\hline 23 & 25 & 24 \\
\hline 30 & 17 & 19 \\
\hline 27 & 23 & 30
\end{array}$$

Sheryl Ezze

Numerade Educator

01:21

Problem 12

The following data represent a simple random sample of $n=5$ from three populations that are known to be normally distributed. Verify that the $F$ -test statistic is 2.599
$$\begin{array}{ccc}
\text { Sample 1 } & \text { Sample 2 } & \text { Sample 3 } \\
\hline 73 & 67 & 72 \\
\hline 82 & 77 & 80 \\
\hline 82 & 66 & 87 \\
\hline 81 & 67 & 77 \\
\hline 97 & 83 & 96
\end{array}$$

Sheryl Ezze

Numerade Educator

01:44

Problem 13

The data in the table represent the number of corn plants in randomly sampled rows (a 17 -foot Eby 5 -inch strip) for various types of plot. An agricultural researcher wants to know whether the mean numbers of plants for each plot type are equal.
$$\begin{array}{lcccccc}
\text { Plot Type } & \multicolumn{5}{c} {\text { Number of Plants }} \\
\hline \text { Sludge plot } & 25 & 27 & 33 & 30 & 28 & 27 \\
\hline \text { Spring disk } & 32 & 30 & 33 & 35 & 34 & 34 \\
\hline \text { No till } & 30 & 26 & 29 & 32 & 25 & 29 \\
\hline
\end{array}$$
(a) Write the null and alternative hypotheses.
(b) State the requirements that must be satisfied to use the one-way ANOVA procedure.
(c) Use the following partial MINITAB output to test the hypothesis of equal means at the $\alpha=0.05$ level of significance.
CAN'T COPY THE TABLE
(d) Shown are side-by-side boxplots of each type of plot. Do these boxplots support the results obtained in part (c)?
CAN'T COPY THE FIGURE
(e) Verify that the $F$ -test statistic is $7.10 .$
(f) Verify the residuals are normally distributed.

Sheryl Ezze

Numerade Educator

01:23

Problem 14

The data in the table represent the number of pods on a random sample of soybean plants for various plot types. An agricultural researcher wants to determine if the mean numbers of pods for each plot type are equal.
$$\begin{array}{llllllllll}
\text { Plot Type } & \multicolumn{5}{c} {\text { Pods }} \\
\hline \text { Liberty } & 32 & 31 & 36 & 35 & 41 & 34 & 39 & 37 & 38 \\
\hline \text { No till } & 34 & 30 & 31 & 27 & 40 & 33 & 37 & 42 & 39 \\
\hline \text { Chisel plowed } & 34 & 37 & 24 & 23 & 32 & 33 & 27 & 34 & 30 \\
\hline
\end{array}$$
(a) Write the null and alternative hypotheses.
(b) State the requirements that must be satisfied to use the oneway ANOVA procedure.
(c) Use the following MINITAB output to test the hypothesis of equal means at the $\alpha=0.05$ level of significance.
CAN'T COPY THE TABLE
(d) Shown are side-by-side boxplots of each type of plot. Do these boxplots support the results obtained in part (c)?
CAN'T COPY THE FIGURE
(e) Verify that the $F$ -test statistic is 3.77
(f) Verify the residuals are normally distributed.

Sheryl Ezze

Numerade Educator

02:03

Problem 15

At a community college, the mathematics department has been experimenting with four different delivery mechanisms for content in their Intermediate Algebra courses. One method is the traditional lecture (method I), the second is a hybrid format in which half the class time is online and the other half is face-to-face (method II), the third is online (method III), and the fourth is an emporium model from which students obtain their lectures and do their work in a lab with an instructor available for assistance (method IV). To assess the effectiveness of the four methods, students in each approach are given a final exam with the results shown next. Do the data suggest that any method has a different mean score from the others?
$$\begin{array}{lllllllllll}
\text { Method I } & 81 & 81 & 85 & 67 & 88 & 72 & 80 & 63 & 62 \\
& 92 & 82 & 49 & 69 & 66 & 74 & 80 & & \\
\hline \text { Method II } & 85 & 53 & 80 & 75 & 64 & 39 & 60 & 61 & 83 \\
& 66 & 75 & 66 & 90 & 93 & & & & \\
\hline \text { Method III } & 81 & 59 & 70 & 70 & 64 & 78 & 75 & 80 & 52 \\
& 45 & 87 & 82 & 79 & & & & & \\
\hline \text { Method IV } & 86 & 90 & 81 & 61 & 84 & 72 & 56 & 68 & 82 \\
& 98 & 79 & 74 & 82 & & & & &
\end{array}$$
(a) Write the null and alternative hypotheses.
(b) State the requirements that must be satisfied to use the oneway ANOVA procedure.
(c) Assuming the requirements stated in part (b) are satisfied, use the following StatCrunch output to test the hypothesis of equal means at the $\alpha=0.05$ level of significance.
CAN'T COPY THE TABLE
(d) Shown are side-by-side boxplots drawn in StatCrunch of cach delivery method. Do these support the results obtained in part (c)?
CAN'T COPY THE FIGURE
(e) Interpret the $P$ -value.
(f) Verify the residuals are normally distributed.

Sheryl Ezze

Numerade Educator

01:30

Problem 16

An obstetrician knew that there were more live births during the week than on weekends. She wanted to determine whether the mean number of births was the same for each of the five days of the week. She randomly selected eight dates for each of the five days of the week and obtained the following data:
$$\begin{array}{ccccc}
\text { Monday } & \text { Tuesday } & \text { Wednesday } & \text { Thursday } & \text { Friday } \\
\hline 10,456 & 11,621 & 11,084 & 11,171 & 11,545 \\
\hline 10,023 & 11,944 & 11,570 & 11,745 & 12,321 \\
\hline 10,691 & 11,045 & 11,346 & 12,023 & 11,749 \\
\hline 10,283 & 12,927 & 11,875 & 12,433 & 12,192 \\
\hline 10,265 & 12,577 & 12,193 & 12,132 & 12,422 \\
\hline 11,189 & 11,753 & 11,593 & 11,903 & 11,627 \\
\hline 11,198 & 12,509 & 11,216 & 11,233 & 11,624 \\
\hline 11,465 & 13,521 & 11,818 & 12,543 & 12,543 \\
\hline
\end{array}$$
(a) Write the null and alternative hypotheses.
(b) State the requirements that must be satisfied to use the oneway ANOVA procedure.
(c) Use the following MINITAB output to test the hypothesis of equal means at the $\alpha=0.01$ level of significance.
CAN'T COPY THE TABLE
(d) Shown are side-by-side boxplots for each weekday. Do these boxplots support the results obtained in part (c)?
CAN'T COPY THE FIGURE

Sheryl Ezze

Numerade Educator

02:16

Problem 17

A stock analyst wondered whether the mean rate of return of financial, energy, and utility stocks differed over the past 5 years. He obtained a simple random sample of eight companies from each of the three sectors and obtained the 5-year rates of return shown in the following table (in percent):
$$\begin{array}{|c|c|c|}
\hline \text { Financial } & \text { Energy } & \text { Utilities } \\
\hline 10.76 & 12.72 & 11.88 \\
\hline 15.05 & 13.91 & 5.86 \\
\hline 17.01 & 6.43 & 13.46 \\
\hline 5.07 & 11.19 & 9.90 \\
\hline 19.50 & 18.79 & 3.95 \\
\hline 8.16 & 20.73 & 3.44 \\
\hline 10.38 & 9.60 & 7.11 \\
\hline 6.75 & 17.40 & 15.70 \\
\hline
\end{array}$$
(a) State the null and alternative hypotheses.
(b) Verify that the requirements to use the one-way ANOVA procedure are satisfied. Normal probability plots indicate that the sample data come from normal populations.
(c) Are the mean rates of return different at the $\alpha=0.05$ level of significance?
(d) Draw boxplots of the three sectors to support the results obtained in part (c).

Alexander Cheng

Numerade Educator

02:02

Problem 18

In an online psychology experiment =sponsored by the University of Mississippi, researchers asked =study participants to respond to various stimuli. Participants were randomly assigned to one of three groups. Subjects in group 1 were in the simple group. They were required to respond as quickly as possible after a stimulus was presented. Subjects in group 2 were in the go/no-go group. These subjects were required to respond to a particular stimulus while disregarding other =stimuli. Finally, subjects in group 3 were in the choice group.
They needed to respond differently, depending on the stimuli presented. Depending on the type of whistle sound, the subject must press a certain button. The reaction time (in seconds) for
each stimulus is presented in the table.
$$\begin{array}{lcc}
\text { Simple } & \text { Go/No Go } & \text { Choice } \\
\hline 0.430 & 0.588 & 0.561 \\
\hline 0.498 & 0.375 & 0.498 \\
\hline 0.480 & 0.409 & 0.519 \\
\hline 0.376 & 0.613 & 0.538 \\
\hline 0.402 & 0.481 & 0.464 \\
\hline 0.329 & 0.355 & 0.725 \\
\hline
\end{array}$$
The researcher wants to determine if the mean reaction times for each stimulus are equal.
(a) State the null and alternative hypotheses.
(b) Verify that the requirements to use the one-way ANOVA procedure are satisfied. Normal probability plots indicate that the sample data come from a normal population.
(c) Test the hypothesis that the mean reaction times for the three stimuli are the same at the $\alpha=0.05$ level of significance.
(d) Draw boxplots of the three stimuli to support the analytic results obtained in part
$\mathbb{B}(c)$.

Sheryl Ezze

Numerade Educator

01:42

Problem 19

To determine if there is gender and/or race discrimination in car buying, Ian Ayres put together a team of fifteen white males, five white females, four black males, and seven black females who were each asked to obtain an initial offer price from the dealer on a certain model car. The 31 individuals were made to appear as similar as possible to account for other variables that may play a role in the offer price of a car. The following data are based on the results in the article and represent the profit on the initial price offered by the dealer.
$$\begin{array}{|c|c|c|c|}
\hline \text { White Male } & \text { Black Male } & \text { White Female } & \text { Black Female } \\
\hline 1300 & 853 & 1241 & 951 & 1899 \\
\hline 646 & 727 & 1824 & 954 & 2053 \\
\hline 951 & 559 & 1616 & 754 & 1943 \\
\hline 794 & 429 & 1537 & 706 & 2168 \\
\hline 661 & 1181 & & 596 & 2325 \\
\hline 824 & 853 & & & 1982 \\
\hline 1038 & 877 & & & 1780 \\
\hline 754 & & & & \\
\hline
\end{array}$$
(a) Ayres wanted to determine if the profit based on the initial offer differed among the four groups. State the null and alternative hypotheses.
(b) A normal probability plot of each group suggests the data come from a population that is normally distributed. Verify the requirement of equal variances is satisfied.
(c) Test the hypothesis stated in part (a).
(d) Draw side-by-side boxplots of the four groups to support the analytic results of part (c).
(e) What do the results of the analysis suggest?
(f) Because the group of black males has a small sample size, the normality requirement is best verified by assessing the normality of the residuals. Verify the normality requirement by drawing a normal probability plot of the residuals.

Sheryl Ezze

Numerade Educator

01:03

Problem 20

The Insurance Institute for Highway Safety conducts experiments in which cars are crashed into a fixed barrier at $40 \mathrm{mph}$. In the institute's $40-\mathrm{mph}$ offset test, $40 \%$ of the total width of each vehicle strikes a barrier on the driver's side. The barrier's deformable face is made of aluminum honeycomb, which makes the forces in the test similar to those involved in a frontal offset crash between two vehicles of the same weight, each going just less than $40 \mathrm{mph}$. You are in the market to buy a family car and you want to know if the mean head injury resulting from this offset crash is the same for large family cars, passenger vans, and midsize utility vehicles. The following data were collected from the institute's study.
CAN'T COPY THE TABLE
The researcher wants to determine if the means for head injury for each class of vehicle are different.
(a) State the null and alternative hypotheses.
(b) Verify that the requirements to use the one-way ANOVA procedure are satisfied. Normal probability plots indicate that the sample data come from normal populations.
(c) Test the hypothesis that the mean head injury for each vehicle type is the same at the $\alpha=0.01$ level of significance.
(d) Draw boxplots of the three vehicle types to support the analytic results obtained in part
$(c)$

Sheryl Ezze

Numerade Educator

01:18

Problem 21

An environmentalist wanted to determine if the mean acidity of rain differed among Alaska, Florida, and Texas. He randomly selected six rain dates at each of the three locations and obtained the following data:
$$\begin{array}{ccc}
\text { Alaska } & \text { Florida } & \text { Texas } \\
\hline 5.41 & 4.87 & 5.46 \\
\hline 5.39 & 5.18 & 6.29 \\
\hline 4.90 & 4.40 & 5.57 \\
\hline 5.14 & 5.12 & 5.15 \\
\hline 4.80 & 4.89 & 5.45 \\
\hline 5.24 & 5.06 & 5.30
\end{array}$$
(a) State the null and alternative hypotheses.
(b) Verify that the requirements to use the one-way ANOVA procedure are satisfied. Normal probability plots indicate that the sample data come from a normal population.
(c) Test the hypothesis that the mean pHs in the rainwater are the same at the $\alpha=0.05$ level of significance.
(d) Draw boxplots of the $\mathrm{pH}$ in rain for the three states to support the results obtained in part (c).

Sheryl Ezze

Numerade Educator

02:15

Problem 22

Researchers Francisco Fuentes and his colleagues wanted to determine the most effective diet for reducing LDL cholesterol, the so-called "bad" cholesterol, among three diets:
(1) a saturated-fat diet, ( 2 ) the Mediterranean diet, and (3) the U.S. National Cholesterol Education Program or NCEP-1 Diet. The participants in the study were shown to have the same levels of LDL cholesterol before the study. Participants were randomly assigned to one of the three treatment groups. Individuals in group 1 reccived the saturated-fat diet, which is $15 \%$ protein, $47 \%$ carbohydrates, and $38 \%$ fat $(20 \% \text { saturated fat, } 12 \%$ monounsaturated fat, and
$6 \%$ polyunsaturated fat). Individuals in group 2 received the Mediterranean diet, which is $47 \%$ carbohydrates, and $38 \%$ fat $(<10 \% \text { saturated fat, } 22 \% \text { monounsaturated fat, and } 6 \%$ polyunsaturated fat). Individuals in group 3 received the NCEP- 1 Diet $(<10 \% \text { saturated fat, } 12 \% \text { monounsaturated fat, and } 6 \%$ polyunsaturated fat). After 28 days, their LDL cholesterol levels were recorded. The data in the following table are based on this study.
$$\begin{array}{|c|c|c|c|cc|}
\hline \text { Saturated Fat } & \multicolumn{2}{|c|} {\text { Mediterranean }} & \multicolumn{2}{|c} {\text { NCEP-1 }} \\
\hline 245 & 218 & 56 & 131 & 125 & 184 \\
\hline 123 & 173 & 78 & 125 & 100 & 116 \\
\hline 166 & 223 & 101 & 160 & 140 & 144 \\
\hline 104 & 177 & 158 & 130 & 151 & 101 \\
\hline 196 & 193 & 145 & 83 & 138 & 135 \\
\hline 300 & 224 & 118 & 263 & 268 & 144 \\
\hline 140 & 149 & 145 & 150 & 75 & 130 \\
\hline 240 & & 211 & & 71 & \\
\hline
\end{array}$$
(a) State the null and alternative hypotheses.
(b) Verify that the requirements to use the one-way ANOVA procedure are satisfied. Normal probability plots indicate that the sample data come from normal populations.
(c) Are the mean LDL cholesterol levels different at the $\alpha=0.05$ level of significance?
(d) Draw boxplots of the LDL cholesterol levels for the three groups to support the analytic results obtained in part (c).

Sheryl Ezze

Numerade Educator

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Problem 23

Treat the results of the Sullivan Statistics Survey as a random sample of adult Americans. The individuals were asked to disclose their political philosophy (conservative, moderate, liberal) as well as the income level they consider to be rich. Load the results of the survey into
a statistical spreadsheet.
(a) If we want to know if there is a difference in the mean income level considered to be rich among the three groups, state the null and alternative hypotheses.
(b) Verify that the requirements to use the one-way ANOVA procedure are satisfied. Normal probability plots indicate the sample data come from normal populations.
(c) Are the mean income levels considered to be rich for the three political philosophies different at the $\alpha=0.5$ level of significance?
(d) Draw boxplots of the income levels of the three political philosophies to support the analytic results obtained in part (c).

Shu Naito

Numerade Educator

02:21

Problem 24

Do people's political philosophy tend to change with age? One technique we may use to answer this question is to see if the mean age of conservatives, moderates, and liberals differ. Load the results of the survey into a statistical spreadsheet and treat the results as a random sample of adult Americans.
(a) State the null and alternative hypotheses.
(b) Verify that the requirements to use the one-way ANOVA procedure are satisfied. Normal probability plots indicate that the sample data come from normal populations.
(c) Are the mean ages for the three political philosophies different at the $\alpha=0.1$ level of significance?
(d) Draw boxplots of the ages of the three political philosophies to support the analytic results obtained in part (c).

Shu Naito

Numerade Educator

01:15

Problem 25

An engineer wants to know if the mean strengths of three different concrete mix designs differ significantly. He randomly selects 9 cylinders that measure
6 inches in diameter and 12 inches in height in which mixture $67-0-301$ is poured, 9 cylinders of mixture $67-0-400,$ and 9 cylinders of mixture $67-0-353 .$ After 28 days, he measures the strength (in pounds per square inch) of the cylinders. The results are presented in the following table:
CAN'T COPY THE TABLE
(a) State the null and alternative hypotheses.
(b) Explain why we cannot use one-way ANOVA to test these hypotheses.

Shu Naito

Numerade Educator

01:13

Problem 26

Researchers (Brian G. Feagan et al., "Erythropoietin with Iron Supplementation to Prevent Allogeneic Blood Transfusion in Total Hip Joint Arthroplasty." Annals of Internal Medicine, Vol. $133,$ No. 11 ) wanted to determine whether epoetin alfa was effective in increasing the hemoglobin concentration in patients undergoing hip arthroplasty. The researchers screened patients for eligibility by performing a complete medical history and physical of the patients. Once eligible patients were identified, the researchers used a computer-generated schedule to assign the patients to the high-dose epoetin group, low-dose epoetin group, or placebo group. The study was double-blind. Based on an analysis of variance, it was determined that there were significant differences in the increase in hemoglobin concentration in the three groups with a $P$ -value less than $0.001 .$ The mean increase in hemoglobin in the high-dose epoetin group was 19.5 grams per liter $(g / L),$ the mean increase in hemoglobin in the low-dose epoctin group was $17.2 \mathrm{g} / \mathrm{L},$ and mean increase in hemoglobin in the placebo group was $1.2 \mathrm{g} / \mathrm{L}$.
(a) Why do you think it was necessary to screen patients for eligibility?
(b) Why was a computer-generated schedule used to assign patients to the various treatment groups?
(c) What does it mean for a study to be double-blind? Why do you think the researchers desired a double-blind study?
(d) Interpret the reported $P$ -value.

Sheryl Ezze

Numerade Educator

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Problem 27

Researchers wanted to determine if the psychological profile of healthy children was different than for children suffering from recurrent abdominal pain (RAP) or recurring headaches. A total of 210 children and adolescents were studied and their psychological profiles were graded according to the Child Behaviour Checklist $4-18$ (CBCL). Children were stratified in two age groups: 4 to 11 years and 12 to 18 years. The results of the study are summarized in the following table:
$$\begin{array}{lccc}
& n & \text { Sample Mean } & \text { Sample Variance } \\
\hline \text { Control group } & 70 & 11.7 & 21.6 \\
\hline \text { RAP } & 70 & 9.0 & 13.0 \\
\hline \text { Headache } & 70 & 12.4 & 8.4 \\
\hline
\end{array}$$
(a) Compute the sample standard deviations for each group.
(b) What sampling method was used for each treatment group? Why?
(c) Use a two sample $t$ -test for independent samples to determine if there is a significant difference in mean CBCL scores between the control group and the RAP group (assume that both samples are simple random samples).
(d) Is it necessary to check the normality assumption to answer part (c)? Explain.
(e) Use the one-way ANOVA procedure with $\alpha=0.05$ to determine if the mean CBCL scores are different for the three treatment groups.
(f) Based on your results from parts (c) and (e), can you determine if there is a significant difference between the mean scores of the RAP group and the headache group? Explain.