Putting It Together: Psychological Profiles Researchers wanted to determine if the

The acronym ANOVA stands for .

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

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

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

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 .

True or False: The F-test statistic is F0 = MST MSE .

In problems 7 and 8, fill in the ANOVA table.Source of Variation Sum of Squares Degrees of Freedom Mean Squares F-Test Statistic Treatment 387 2 Error 8042 27 Total

In problems 7 and 8, fill in the ANOVA table.Source of Variation Sum of Squares Degrees of Freedom Mean Squares F-Test Statistic Treatment 2814 3 Error 4915 36 Total

In Problems 9 and 10, determine the F-test statistic based on the given summary statistics. [Hint: x = ni xi ni .] Population Sample Size Sample Mean Sample Variance 1 10 40 48 2 10 42 31 3 10 44 25

In Problems 9 and 10, determine the F-test statistic based on the given summary statistics. [Hint: x = ni xi ni .]Population Sample Size Sample Mean Sample Variance 1 15 105 34 2 15 110 40 3 15 108 30 4 15 90 38

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. Sample 1 Sample 2 Sample 3 28 22 25 23 25 24 30 17 19 27 23 30

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. Sample 1 Sample 2 Sample 3 73 67 72 82 77 80 82 66 87 81 67 77 97 83 96

Corn Production The data in the table represent the number of corn plants in randomly sampled rows (a 17-foot by 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. Plot Type Number of Plants Sludge plot 25 27 33 30 28 27 Spring disk 32 30 33 35 34 34 No till 30 26 29 32 25 29 Source: Andrew Dieter and Brad Schmidgall, Joliet Junior College (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 a = 0.05 level of significance. One-way ANOVA: Sludge Plot, Spring Disk, No Till DF 2 15 17 Source Factor Error Total MS 42.06 5.92 SS 84.11 88.83 172.94 F 7.10 P 0.007 (d) Shown are side-by-side boxplots of each type of plot. Do these boxplots support the results obtained in part (c)? 25 Sludge Plot Number of Plants Plot Type 30 35 Spring Disk No Till (e) Verify that the F-test statistic is 7.10. (f ) Verify the residuals are normally distributed.

Corn Production The data in the table represent the number of corn plants in randomly sampled rows (a 17-foot by 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. Plot Type Number of Plants Sludge plot 25 27 33 30 28 27 Spring disk 32 30 33 35 34 34 No till 30 26 29 32 25 29 Source: Andrew Dieter and Brad Schmidgall, Joliet Junior College (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 a = 0.05 level of significance. One-way ANOVA: Sludge Plot, Spring Disk, No Till DF 2 15 17 Source Factor Error Total MS 42.06 5.92 SS 84.11 88.83 172.94 F 7.10 P 0.007 (d) Shown are side-by-side boxplots of each type of plot. Do these boxplots support the results obtained in part (c)? 25 Sludge Plot Number of Plants Plot Type 30 35 Spring Disk No Till (e) Verify that the F-test statistic is 7.10. (f ) Verify the residuals are normally distributed. (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 a = 0.05 level of significance. One-way ANOVA: Liberty, No Till Chisel Plowed DF 2 24 26 Source Factor Error Total MS 74.5 19.8 SS 149.0 474.7 623.6 F 3.77 P 0.038 (d) Shown are side-by-side boxplots of each type of plot. Do these boxplots support the results obtained in part (c)? 20 Liberty Number of Plants Plot Type 30 40 No Till Chisel Plowed (e) Verify that the F-test statistic is 3.77. (f ) Verify the residuals are normally distributed.

Which Delivery Method Is Best? 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? Method I 81 81 85 67 88 72 80 63 62 92 82 49 69 66 74 80 Method II 85 53 80 75 64 39 60 61 83 66 75 66 90 93 Method III 81 59 70 70 64 78 75 80 52 45 87 82 79 Method IV 86 90 81 61 84 72 56 68 82 98 79 74 82 (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 a = 0.05 level of significance. (d) Shown are side-by-side boxplots drawn in StatCrunch of each delivery method. Do these support the results obtained in part (c)? (e) Interpret the P-value. (f ) Verify the residuals are normally distributed.

Births by Day of Week 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: Monday Tuesday Wednesday Thursday Friday 10,456 11,621 11,084 11,171 11,545 10,023 11,944 11,570 11,745 12,321 10,691 11,045 11,346 12,023 11,749 10,283 12,927 11,875 12,433 12,192 10,265 12,577 12,193 12,132 12,422 11,189 11,753 11,593 11,903 11,627 11,198 12,509 11,216 11,233 11,624 11,465 13,521 11,818 12,543 12,543 Source: National Center for Health Statistics (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 a = 0.01 level of significance. One-way ANOVA: Mon, Tues, Wed, Thurs, Fri DF 4 35 39 Source Factor Error Total MS 2876908 293451 SS 11507633 10270781 21778414 F 9.80 P 0.000 (d) Shown are side-by-side boxplots for each weekday. Do these boxplots support the results obtained in part (c)? Wednesday Tuesday Monday Number of Births Day Thursday Friday 10,000 11,000 12,000 13,000

Rates of Return 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): Financial Energy Utilities 10.76 12.72 11.88 15.05 13.91 5.86 17.01 6.43 13.46 5.07 11.19 9.90 19.50 18.79 3.95 8.16 20.73 3.44 10.38 9.60 7.11 6.75 17.40 15.70 Source: Morningstar.com (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 a = 0.05 level of significance? (d) Draw boxplots of the three sectors to support the results obtained in part (c).

Reaction Time 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. Simple Go/No Go Choice 0.430 0.588 0.561 0.498 0.375 0.498 0.480 0.409 0.519 0.376 0.613 0.538 0.402 0.481 0.464 0.329 0.355 0.725 Source: PsychExperiments; The University of Mississippi; www.olemiss.edu/psychexps/ 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 a = 0.05 level of significance. (d) Draw boxplots of the three stimuli to support the analytic results obtained in part (c).

Car-Buying Discrimination 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. White Male Black Male White Female Black Female 1300 853 1241 951 1899 646 727 1824 954 2053 951 559 1616 754 1943 794 429 1537 706 2168 661 1181 596 2325 824 853 1982 1038 877 1780 754 Source: Ian Ayres. Fair Driving: Gender and Race Discrimination in Retail Car Negotiations. Harvard Law Review. Vol. 104, No. 4, Feb. 1991.(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

Crash Data The Insurance Institute for Highway Safety conducts experiments in which cars are crashed into a fixed barrier at 40 mph. In the institutes 40-mph offset test, 40% of the total width of each vehicle strikes a barrier on the drivers side. The barriers 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 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 institutes study. Large Family Cars Head Injury (hic) Hyundai XG300 264 Ford Taurus 134 Buick LeSabre 409 Chevrolet Impala 530 Chrysler 300 149 Pontiac Grand Prix 627 Toyota Avalon 166 Passenger Vans Head Injury (hic) Toyota Sienna 148 Honda Odyssey 238 Ford Freestar 340 Mazda MPV 693 Chevrolet Uplander 550 Nissan Quest 470 Kia Sedona 322 Midsize Utility Vehicles Head Injury (hic) Honda Pilot 225 Toyota 4Runner 216 Mitsubishi Endeavor 186 Nissan Murano 307 Ford Explorer 353 Kia Sorento 552 Chevy Trailblazer 397 Source: Insurance Institute for Highway Safety 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 a = 0.01 level of significance. (d) Draw boxplots of the three vehicle types to support the analytic results obtained in part (c).

pH in Rain 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: Alaska Florida Texas 5.41 4.87 5.46 5.39 5.18 6.29 4.90 4.40 5.57 5.14 5.12 5.15 4.80 4.89 5.45 5.24 5.06 5.30 Source: National Atmospheric Deposition Program (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 a = 0.05 level of significance. (d) Draw boxplots of the pH in rain for the three states to support the results obtained in part (c).

Lower Your Cholesterol 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 received the saturated-fat diet, which is 15% protein, 47% carbohydrates, and 38% fat (20% saturated fat, 12% monounsaturated fat, and 6% polyunsaturated fat). Individuals in group 2 received the Mediterranean diet, which is 47% carbohydrates, and 38% fat (610% saturated fat, 22% monounsaturated fat, and 6% polyunsaturated fat). Individuals in group 3 received the NCEP-1 Diet (610% saturated fat, 12% 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. Saturated Fat Mediterranean NCEP-1 245 218 56 131 125 184 123 173 78 125 100 116 166 223 101 160 140 144 104 177 158 130 151 101 196 193 145 83 138 135 300 224 118 263 268 144 140 149 145 150 75 130 240 211 71 (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 a = 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).

Sullivan Statistics Survey: What Is Rich? 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 a = 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).

Sullivan Statistics Survey: Age and Politics Do peoples 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 a = 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).

Concrete Strength 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: Mixture 67-0-301 Mixture 67-0-400 Mixture 67-0-353 3960 4090 4070 4120 4150 3820 4040 3830 4330 4640 3820 3750 3780 3940 4620 4190 4010 3990 3890 4080 3730 3850 4150 4320 3990 4890 4190 (a) State the null and alternative hypotheses. (b) Explain why we cannot use one-way ANOVA to test these hypotheses.

Analyzing Journal Article Results 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 epoetin group was 17.2 g/L, and mean increase in hemoglobin in the placebo group was 1.2 g/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.

Putting It Together: Psychological Profiles 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 418 (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: n Sample Mean Sample Variance Control group 70 11.7 21.6 RAP 70 9.0 13.0 Headache 70 12.4 8.4 Source: Galli, et al. Headache and Recurrent Abdominal Pain: A Controlled Study by Means of the Child Behaviour Checklist (CBCL). Cephalalgia 27, 211219, 2007 (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 a = 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.

28. What are the requirements to perform a one-way ANOVA? Is the test robust?

What is the mean square due to treatment estimate of s2 ? What is the mean square due to error estimate of s2 ?

Why does a large value of the F statistic provide evidence against the null hypothesis H0: m1 = m2 = g = mk?

In a one-way ANOVA, explain what it means to reject the statement in the null hypothesis if three treatment groups are being compared.