Unit 3 Section 1 Exercises

1. In this section we discuss comparing means from a number of populations. Can you think of a situation where one might wish to compare variances among populations? compare upper quartiles among populations?
2. List the three assumptions under which analysis of variance procedures are developed (see page 385).
3. The sample variance of the means of the multiple populations is given as. Is this the within means variance S_W² or the between means variance, S_B²?
4. The F-test for equality of means from multiple populations if formed from the ratio of two variance estimates. Which of the test statistics below is the correct ratio?
1. 
2. 
5. One column of the analysis of variance table for a completely randomized design consists of some sums of squares. Below are two column lists. In the first column are the names of the sums of squares and in the second column are the equations for the sums of squares. Match the equations with their correct names.

TSS (total sums of squares)	a.
SSW (within sample sums of squares)	b.
SSB (between sample sums of squares)	c.

 

6. Suppose you have performed an experiment using a completely randomized design to compare t=5 treatments. Each treatment has been replicated ni=4 times. Complete the table below with the appropriate degrees of freedom.

Source	Degrees of Freedom
Between samples
Within samples
Totals

7. Using the degrees of freedom from the table above and Table 8 in the book, what is the critical value for the F test at a type I error rate of 5%?
8. The linear model typically used to analyze a one-way classification study (completely randomized design) has the form: . Describe in your own words what each term of the model describes about the data. For example, the m term describes the overall average for the data.
9. Can you explain why the null hypothesis that all the a_i terms are equal to zero is equivalent to the hypothesis that all treatment means are the same.
10. Below are listed typical responses that would not be expected to have homogeneous variances. Beside each response list the transformation typically used prior to the analysis of variance in an attempt to produce more homogeneous variances.

Response	Transformation
Counts between 0 and 10
Counts above 10
Proportion
Percentage
Rate

11. How does the hypothesis of the Kruskal-Wallis Test differ from the hypothesis tested with the usual analysis of variance procedures?

Problem 8.17 in Ott and Longnecker. Researchers record the yields of corn, in bushels per plot, for four different varieties of corn, A, B, C, and D. In a controlled greenhouse experiment, the researchers randomly assign each variety to eight of 32 plots available for the study. The yields are listed here.

 A   B   C   D
2.5 3.6 4.3 2.8
3.6 3.9 4.4 2.9
2.8 4.1 4.5 3.1
2.7 4.3 4.1 2.4
3.1 2.9 3.5 3.2
3.4 3.5 3.4 2.5
2.9 3.8 3.2 3.6
3.5 3.7 4.6 2.7

1. Write the linear model for these data.
2. Perform a one-way analysis of variance. Test the hypothesis of no differences in average yields at a type I error rate of 5%.
3. Test the hypothesis of equal variances. Do these data need to be transformed? If so, what transformation would you use?
4. For each variety, compute the sample mean and subtract this mean from each observation (i.e. center each variety). Combine all these "residuals" into on data set and test for normality. Is there any indication that these data are not normal.
5. Perform the Kruskal-Wallis test on these data. How do the results compare to the analysis of variance procedure?

Problem 8.30, page 424 of Ott and Longnecker. A small corporation make insulation shields for electrical wires using three different types of machines. The corporation wants to evaluate the variation in the inside diameter dimension of the shields produced by the machines. A quality engineer at the corporation randomly selects shields produced by each of the machines and records the inside diameters of each shield (in millimeters). She wants to determine whether the means and standard deviations of the three machines differ. The data are given below.

Machine_C  Machine_B  Machine_A
 29.7        8.7         18.1
 18.7       56.8          2.4
 16.5        4.4          2.7
 63.7        8.3          7.5
 18.9        5.8         11.0
107.2         .            .
 19.7         .            .
 93.4         .            .
 21.6         .            .
 17.8         .            .

1. Write the linear model for these data.
2. Perform a one-way analysis of variance. Test the hypothesis of no differences in average shield diameters at a type I error rate of 5%.
3. Test the hypothesis of equal variances. Do these data need to be transformed? If so, what transformation would you use?
4. For each machine, compute the sample mean and subtract this mean from each observation (i.e. center each machine's data). Combine all these "residuals" into on data set and test for normality. Is there any indication that these data are not normal.
5. Perform the Kruskal-Wallis test on these data. How do the results compare to the analysis of variance procedure?

From problem 9.24,page 464 in Ott and Longnecker. Researchers conducted a study of the effects of three heartworm drugs on the fat content of the shoulder muscles in Labrador retrievers. Twenty (20) dogs were randomly divided into four treatment groups of five (5) dogs each. Dogs in group A were assigned to the untreated control. The remaining three groups identify which of the heartworm drug all dogs in the group received. At the end of two years of treatment, the percentage fat content of the shoulder muscles was determined and is given here.

 A    B    C    D
2.64 3.24 2.90 2.91
2.56 3.71 3.02 2.89
3.30 2.95 3.78 3.21
2.19 3.01 2.96 2.89
2.45 3.08 2.87 2.68

1. Write the linear model for these data.
2. Perform a one-way analysis of variance. Test the hypothesis of no differences in average fat content at a type I error rate of 5%.
3. Test the hypothesis of equal variances. Do these data need to be transformed? If so, what transformation would you use?
4. For each group, compute the sample mean and subtract this mean from each observation (i.e. center each group's data). Combine all these "residuals" into on data set and test for normality. Is there any indication that these data are not normal.
5. Perform the Kruskal-Wallis test on these data. How do the results compare to the analysis of variance procedure?

A large advertising firm specializes in creating television commercials for children's products. A study is designed to test whether there are large differences in attention span for children in three different age groups (Group A 5-6 years, Group B 7-8 years, Group C 9-10 years). Ten children for each group are selected at random from a large population of children and asked to watch a 60-second breakfast cereal commercial. Researchers record their attention span during the viewing of the commercial. Times (in seconds) are given below.

 A  B  C
19 19 37
36 35  6
40 22 28
30 28  4
 4  1 32
10 27 16
30 27  8
 5 16 41
34  3 29
21 18 18

1. Write the linear model for these data.
2. Perform a one-way analysis of variance. Test the hypothesis of no differences in average attention time at a type I error rate of 5%.
3. Test the hypothesis of equal variances. Do these data need to be transformed? If so, what transformation would you use?
4. For each group, compute the sample mean and subtract this mean from each observation (i.e. center each group's data). Combine all these "residuals" into on data set and test for normality. Is there any indication that these data are not normal.
5. Perform the Kruskal-Wallis test on these data. How do the results compare to the analysis of variance procedure.