Analysis of Variance (57 problems)

. In a bumper test, three types of autos were deliberately crashed into a barrier at 5 mph, and the resulting damage (in dollars) was estimated. Five test vehicles of each type were crashed, with the results shown below. Research question: Are the mean crash damages the same for these three vehicles?

An experiment was conducted to compare the effectiveness of three training programs, A, B, and C, in training assemblers of a piece of electronic equipment. Fifteen employees were randomly assigned, 5 each, to the three programs. After the completion of the courses, each person was required to assemble four pieces of equipment, and the average length of time required to complete the assembly was recorded. Owing to resignation from the company, only 4 employees completed the program A and only 3 completed the program B while all 5 employees completed the program C. The result is as follows:

Do the data provides sufficient evidence to indicate a difference in mean assembly times for people trained by the three programs at a 95% confidence level?

Professor Brunner had students in his Marketing class rate his performance as Excellent, Good, Fair or Poor. A graduate student collected the ratings and assured the students that Professor Brunner would not receive them until after the course grades had been submitted. The rating (i.e. the treatment) a student gave the professor was matched with his/her course grade, which could range from 0 to 100. The data is as follows:

Course Grades

Is there a difference in the mean scores of the students in each of the four rating categories at a 99% confidence level?

The manager of a computer software company wishes to study the number of hours senior executives spend at their desktop computers by type of industry. The manager selected a sample of five executives from each of three industries. At the .05 significance level, can she conclude there is a difference in the mean number of hours spent per week by industry?

BIRTH RATE ANALYSIS

It is said that first born children tend to develop language skills faster than their younger siblings. One possible explanation for this phenomenon is that first-borns have undivided attention from their parents. If this explanation is correct, then it is also reasonable that twins should show slower language development that since children and that triplets should be even slower. Davis (1937) found that exactly this result. The following hypothetical data demonstrate the relationship. The dependent variable is a measure of language skill at age 3 for each child. Do the data indicate any differences? Test with a = .05.

To test whether memory changes with age, a researcher conducts an experiment in which there are four groups of six subjects each. The groups differ according to the age of the subjects. In group 1, the subjects are each 30 years old; group 2, 40 years old; group 3, 50 years olf; and group 4, 60 years old. Assume that subjects are all in good health and that the groups are matched on other important variable such as years of education, IQ, sex, motivation, etc. Each subject is shown a series of nonsense syllables (a meaningless combination of three letters such as DAF or FUM) at a rate of one syllable every 4 seconds. The series is shown twice after which the subjects are asked to write down as many of the syllables as they can remember. The number of syllables remembered by each subject is shown below.

__________________________________________________________________

30 Years 40 Years 50 Years 60 Years

Old Old Old Old_______

14 12 17 13

13 15 14 10

15 16 14 7

17 11 9 8

12 12 13 6

10 18 15 9

Use the analysis of variance with a = 0.05 to determine if age has an effect on memory.

A medical researcher wishes to test the effects of 2 diets and the time of day on the sodium level in person’s blood. Eight people are randomly selected, and two are randomly assigned to each of the four groups. Analyze the data shown in the tables here, using a two-way ANOVA at a=0.05. The sodium content is measured in milliequivalents per liter.

Diet Type I II

Time

8:00 am 135 145 138 141

8:00 PM 155 162 171 191

A company sells 3 items: swimming pools, spas, saunas. Owner decides to see whether age of the sales rep and the type of item affect monthly sales. At a=0.05, analyze the data shown, using a two-way ANOVA. Sales are given in hundreds of dollars for a randomly selected month, and five salespeople were selected for each group.

A medical researcher wishes to test the effects of 2 diets and the time of day on the sodium level in person’s blood. Eight people are randomly selected, and two are randomly assigned to each of the four groups. Analyze the data shown in the tables here, using a two-way ANOVA at a=0.05. The sodium content is measured in milliequivalents per liter.

A company sells 3 items: swimming pools, spas, saunas. Owner decides to see whether age of the sales rep and the type of item affect monthly sales. At a=0.05, analyze the data shown, using a two-way ANOVA. Sales are given in hundreds of dollars for a randomly selected month, and five salespeople were selected for each group.

Two programs in outdoor survival are available for army recruits. One lasts one week and the other lasts 2 weeks. The officer wishes to test the effectiveness of the programs and see whether there are any gender differences. Six subjects are randomly assigned to each of the programs according to gender. After completing the program, each is given a written test on his or her knowledge of survival skills. The test consists of 100 questions. The scores of the groups are shown here. Use a=0.10 and analyze the data, using a two-ways ANOVA.

DURATION

One Week Two Weeks

GENDER

Female 86, 92, 87, 88, 78, 95 78, 62, 56, 54, 65, 63

Male 52, 67, 53, 42, 68, 71 85, 94, 82, 84, 78, 91

Use the Excel display, which results from the scores listed in the accompanying table. The sample data are SAT scores on the verbal and math portions of SAT-I and are based on reported statistics from the College Board. The “Columns” variable is SAT (verbal/math) and the “Sample” variable is gender.

Test the null hypothesis that SAT scores are not affected by an interaction between gender and test (verbal/math). What do you conclude?

Car Fuel Consumption. The following table lists highway fuel consumption amounts (in mi/gal) from Data Set 22 in Appendix B. Assume that fuel consumption amounts are not affected by an interaction between the type of transmission (manual of automatic) and the number of cylinders. Are fuel consumption amounts affected by the type of transmission? Are fuel consumption amounts affected by the number of cylinders?

Two programs in outdoor survival are available for army recruits. One lasts one week and the other lasts 2 weeks. The officer wishes to test the effectiveness of the programs and see whether there are any gender differences. Six subjects are randomly assigned to each of the programs according to gender. After completing the program, each is given a written test on his or her knowledge of survival skills. The test consists of 100 questions. The scores of the groups are shown here. Use a=0.10 and analyze the data, using a two-way ANOVA.

In a two-way ANOVA experiment, factor A is operating on 3 levels, factor B is operating on 2 levels, and there are 2 replications per cell. If MSA/MSE = 5.35, MSB/MSE = 5.72, and MSAB/MSE = 6.75, and using the 0.05 level of significance, what conclusions would be reached regarding the respective null hypotheses for this experiment?

Given the following data for a two-way ANOVA, identify the sets of null and alternative hypotheses, then use the 0.05 level in testing each null hypotheses.

Use a repeated measures analysis of variance with a = .05 to determine whether these data are sufficient to demonstrate significant differences between the treatments.