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Chapter 11: Problem 1

Suppose you wish to compare the means of six populations based on independent random samples, each of which contains 10 observations. Insert, in an ANOVA table, the sources of variation and their respective degrees of freedom.

Short Answer

Expert verified
Answer: In the given exercise, there are 5 degrees of freedom for the Between groups category and 54 degrees of freedom for the Within groups category in the ANOVA table.

Step by step solution

01

Identifying the number of groups and total number of observations

In this problem, we have six populations (or groups) and 10 observations in each group. Therefore, we have a total of 60 observations.
02

Calculating the degrees of freedom for Between groups

To calculate the degrees of freedom for Between groups, we use the formula: Degrees of freedom (Between) = Number of groups - 1 In this case, we have 6 populations: Degrees of freedom (Between) = 6 - 1 = 5
03

Calculating the degrees of freedom for Within groups

To calculate the degrees of freedom for Within groups, we use the formula: Degrees of freedom (Within) = Total number of observations - Number of groups In this case, we have a total of 60 observations and 6 populations: Degrees of freedom (Within) = 60 - 6 = 54
04

Creating the ANOVA table

Now we can insert the sources of variation, their respective degrees of freedom, and sum of squares (which were not given in this problem) into an ANOVA table: | Source of Variation | Degrees of Freedom | Sum of Squares | |---------------------|--------------------|----------------| | Between groups | 5 | - | | Within groups | 54 | - | | Total | 59 | - | Note: The sum of squares values were not provided in the exercise, so they have been left as a placeholder "-".

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Most popular questions from this chapter

An experiment was conducted to compare the effectiveness of three training programs, \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C},\) in training assemblers of a piece of electronic equipment. Fifteen employees were randomly assigned, five each, to the three programs. After completion of the courses, each person was required to assemble four pieces of the equipment, and the average length of time required to complete the assembly was recorded. Several of the employees resigned during the course of the program; the remainder were evaluated, producing the data shown in the accompanying table. Use the MINITAB printout to answer the questions. $$ \begin{array}{lllll} \text { Training Program } & {\text { Average Assembly Time (min) }} \\ \hline \text { A } && 59 & 64 & 57 & 62 \\ \text { B } && 52 & 58 & 54 & \\ \text { C } && 58 & 65 & 71 & 63 & 64 \end{array} $$ a. Do the data provide sufficient evidence to indicate a difference in mean assembly times for people trained by the three programs? Give the \(p\) -value for the test and interpret its value. b. Find a \(99 \%\) confidence interval for the difference in mean assembly times between persons trained by programs \(\mathrm{A}\) and \(\mathrm{B}\) c. Find a \(99 \%\) confidence interval for the mean assembly times for persons trained in program A. d. Do you think the data will satisfy (approximately) the assumption that they have been selected from normal populations? Why?

Four chemical plants, producing the same product and owned by the same company, discharge effluents into streams in the vicinity of their locations. To check on the extent of the pollution created by the effluents and to determine whether this varies from plant to plant, the company collected random samples of liquid waste, five specimens for each of the four plants. The data are shown in the table: $$ \begin{array}{llllll} \text { Plant } & {\text { Polluting Effluents }} {\text { (Ib/gal of waste) }} \\ \hline \text { A } && 1.65 & 1.72 & 1.50 & 1.37 & 1.60 \\ \text { B } & &1.70 & 1.85 & 1.46 & 2.05 & 1.80 \\ \text { C } && 1.40 & 1.75 & 1.38 & 1.65 & 1.55 \\ \text { D } && 2.10 & 1.95 & 1.65 & 1.88 & 2.00 \end{array} $$ a. Do the data provide sufficient evidence to indicate a difference in the mean amounts of effluents discharged by the four plants? b. If the maximum mean discharge of effluents is 1.5 lb/gal, do the data provide sufficient evidence to indicate that the limit is exceeded at plant \(\mathrm{A} ?\) c. Estimate the difference in the mean discharge of effluents between plants \(\mathrm{A}\) and \(\mathrm{D},\) using a \(95 \%\) confidence interval.

Water samples were taken at four different locations in a river to determine whether the quantity of dissolved oxygen, a measure of water pollution, varied from one location to another. Locations 1 and 2 were selected above an industrial plant, one near the shore and the other in midstream; location 3 was adjacent to the industrial water discharge for the plant; and location 4 was slightly downriver in midstream. Five water specimens were randomly selected at each location, but one specimen, corresponding to location \(4,\) was lost in the laboratory. The data and a MINITAB analysis of variance computer printout are provided here (the greater the pollution, the lower the dissolved oxygen readings). $$ \begin{array}{llllll} \text { Location } && {\text { Mean Dissolved }} {\text { Oxygen Content }} \\\ \hline 1 &&& 5.9 & 6.1 & 6.3 & 6.1 & 6.0 \\ 2 &&& 6.3 & 6.6 & 6.4 & 6.4 & 6.5 \\ 3 &&& 4.8 & 4.3 & 5.0 & 4.7 & 5.1 \\ 4 &&& 6.0 & 6.2 & 6.1 & 5.8 & \end{array} $$ a. Do the data provide sufficient evidence to indicate a difference in the mean dissolved oxygen contents for the four locations? b. Compare the mean dissolved oxygen content in midstream above the plant with the mean content adjacent to the plant (location 2 versus location 3 ). Use a \(95 \%\) confidence interval.

The partially completed ANOVA table for a randomized block design is presented here: $$ \begin{array}{lcl} \text { Source } & d f & \text { SS } & \text { MS } \quad F \\ \hline \text { Treatments } & 4 & 14.2 & \\ \text { Blocks } & & 18.9 & \\ \text { Error } & 24 & & \\ \hline \text { Total } & 34 & 41.9 & \end{array} $$ a. How many blocks are involved in the design? b. How many observations are in each treatment total? c. How many observations are in each block total? d. Fill in the blanks in the ANOVA table. e. Do the data present sufficient evidence to indicate differences among the treatment means? Test using \(\alpha=.05\) f. Do the data present sufficient evidence to indicate differences among the block means? Test using \(\alpha=.05\)

A nationa home builder wants to compare the prices per 1,000 board feet of standard or better grade Douglas fir framing lumber. He randomly selects five suppliers in each of the four states where the builder is planning to begin construction. The prices are given in the table. $$ \begin{array}{rrrr} && {\text { State }} \\ \hline 1 & 2 & 3 & 4 \\ \hline \$ 241 & \$ 216 & \$ 230 & \$ 245 \\ 235 & 220 & 225 & 250 \\ 238 & 205 & 235 & 238 \\ 247 & 213 & 228 & 255 \\ 250 & 220 & 240 & 255 \end{array} $$ a. What type of experimental design has been used? b. Construct the analysis of variance table for this data. c. Do the data provide sufficient evidence to indicate that the average price per 1000 board feet of Douglas fir differs among the four states? Test using \(\alpha=.05\)
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