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# Analysis of Variance (ANOVA)

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#### Solution Summary

A diaper company is considering 3 different filler materials for their disposable diapers. Eight diapers were tested with each of the 3 filler materials and 24 toddlers were randomly given a diaper to wear. As the child played, fluid was injected into the diaper every 10 minutes until the product failed(leaked). The amount of fluid (in grams) at the time of failure was recorded for each diaper. The data are shown below.

Material 1 Material 2 Material 3
791 809 828
789 818 814
796 803 855
802 781 844
810 813 847
790 808 848
800 805 836
790 811 873

a) What is the response variable and what is the factor?
b) How many levels of the factor are being studied?
c) Is there any difference in the average amount of fluid the diaper can hold using the 3 different filler materials? If so, which ones are different?
d) What is your recommendation to the company and why?

14.3
Grading homework is a real problem. It takes an enormous amount of time and many students do not do a very good job or copy answers from other students or the back of the book. A teacher of elementary statistics decided to conduct a study to determine what effect grading homework had on her students' exam scores. She taught 3 sections of Elementary Statistics and randomly assigned each class one of three conditions: (1) no homework given, (2) homework given but not collected, and (3) homework given, collected, and graded. After the first exam, she collected the data (exam scores). They are shown below.

No Homework, Homework,
Homework Not Collected Collected

69 73 83
69 63 97
92 68 72
84 79 79
79 57 84
84 68 76
76 72 91
63 74 76
76 49 83
82 84 88
89 79 91
72 71 96
72 80 68
65 74 99
73 71 89
47 63 80
92 88 79
71 83 91
83 89 83
81 82 83
92 69 76
80 92 90
64 79 79
72 81 67
84 76 86
79 81 86
74 81 82
81 75 84
a) What is the response variable and what is the factor?
b) How many levels of the factor are being studied?
c) Is there any difference in the average exam scores using the 3 different approaches to homework? If so, which ones are different?
d) What is your recommendation to the teacher and why?

14.5
The sports industry is a large and competitive market. A manufacturer of golf balls is considering 4 new ball designs. A sample of 36 balls from each of the 4 models is tested and the distance the balls carry (in yards) is recorded. The balls are hit by a machine. The data is in data file BALLDESN.xls (attached Excel file)

a) What is the response variable and what is the factor?
b) How many levels of the factor are being studied?
c) Is there any difference in the average distance the ball carries?
d) What is your recommendation to the golf manufacturer and why?

17.1
The administration of a university has been using the following distribution to classify the ages of their students:

Age Group Estimated % of Student Population

Less than 18 2.7
18-19 29.9
20-24 53.4
Older than 24 14

A recent student survey provided the following data on age of students:

Age Group Frequency

Less than 18 6
18-19 118
20-24 102
Older than 24 26

a) Set up a table that compares the expected and observed frequencies for each group.
b) Based on the table, do you think that the data represent the estimated distribution?
c) Set up the hypothesis for the chi-square goodness of fit test.
d) Perform the goodness of fit test at the 0.05 level of significance.
e) Based on the chi-square test, is the estimated age distribution that the university is using correct?

17.2
As part of a survey on the use of Office Suites Software, the company doing the polling wanted to know whether its population was uniformly distributed over the following age distribution: under 25, 25 to 44, 45 and up. The company looked at the data it had collected so far and found the following distribution:

Age Group Number of Respondents

Under 25 73
25-44 61
45 and up 66
Total 200

a) Based on the data, do you think that the respondents are uniformly distributed over the age categories?
b) Set up the hypothesis to test whether the data are uniformly distributed over the age categories.
c) Find the expected frequency distribution and perform the chi-square goodness of fit test.
d) At the 0.05 level of significance, would you say that they respondents were uniformly distributed over the age groups?

T F 1. ANOVA is an abbreviation for analysis of variance.

T F 2. Factors are experimental variables that can be used to differentiate groups or populations from one another.

T F 3. A level contains a collection of possible experimental factors being used to distinguish groups or populations.

4. The chi-square test compares the __________ frequency distribution of the data to the frequency distribution that would be __________ if the null hypothesis were __________.
a. observed, expected, true
b. observed, expected, false
c. expected, observed, true
d. expected, observed, false

5. At an international student organization function, the president of the
organization believed that the students who were present consisted of 10 percent Africans, 25 percent Indians, 40 percent Asians, and 25 percent Americans.
The students present consisted of 25 Africans, 15 Indians, 80 Asians, and 20 Americans. At alpha = 0.05 , test the president's belief.

(1) Set up the table that compares the expected and observed frequencies for each group.
(2) Set up the hypotheses for the chi-square goodness of fit test.
(3) Perform the goodness of fit test at the 0.05 level of significance; alpha =0.05.

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