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Exam 13: Experimental Design and Analysis of Variance

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Question 1

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In the analysis of variance procedure (ANOVA) , factor refers to _____.

A) the dependent variable
B) the independent variable
C) different levels of a treatment
D) the critical value of F

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Question 2

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Random samples of individuals from three different cities were asked how much time they spend per day watching television. The results (in minutes) for the three groups are shown below. $\begin{array} { c c c } \text { City I } & \text { City II } & \text { City III } \\260 & 178 & 211 \\280 & 190 & 190 \\240 & 220 & 250 \\260 & 240 & \\300 & &\end{array}$ At α = .05, use Excel to test to see if there is a significant difference in the averages of the three groups.

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​ \(\begin{array}{|l|l|l|l|l|l|l|l|} \hline & \text { A } & \text { B } & \text { C } & \text { D }& \text { E } & \text { F } & \text { G } \\ \hline \mathbf{1} & \text { Observation } & \text { City I } & \text { City II } & \text { City III } \\ \hline 2 & 1 & 260 & 178 & 211 \\ \hline 3 & 2 & 280 & 190 & 190 \\ \hline 4 & 3 & 240 & 220 & 250 \\ \hline 5 & 4 & 260 & 240 & \\ \hline 6 & 5 & 300 & &\\\hline 7 & \\ \hline 8 & \text { Anova: Single Fac tor } \\ \hline 9 & \\ \hline 10 & \text { SUMMARY } \\\hline \mathbf{1 1} & \text { Groups } & \text { Count } & \text { Sum } & \text { Average } & \text { Variance } \\ \hline \mathbf{1 2} & \text { City I } & 5 & 1340 & 268 & 520 \\ \hline \mathbf{1 3} & \text { City II } & 4 & 828 & 207 & 796 \\ \hline \mathbf{1 4} & \text { City III } & 3 & 651 & 217 & 927 \\\hline15 & \\ \hline 16 & \text { ANOVA }\\ \hline 17 & \text { Source of Variation } & S S & d f & M S & F & \text { P-value } & F_{\text {crit }} \\ \hline 18 & \text { Between Groups } & 9552.92 & 2 & 4776.458 & 6.79977 & 0.01587 & 4.25649 \\ \hline 19 & \text { Within Groups } & 6322.00 & 9 & 702.444 & & & \\\hline 20 & & & \\ \hline 21 & \text { Total } & 15874.92 & 11 \\\hline \end{array}\) Reject H_0, conclude that there is a significant difference in the averages of the three groups

Question 3

Multiple Choice

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An experimental design where the experimental units are randomly assigned to the treatments is known as _____.

The critical F value with 6 numerator and 60 denominator degrees of freedom at α = .05 is _____.

Random samples were selected from three populations. The data obtained are shown below. $\begin{array} { c c c } \text { Treatment 1 } & \text { Treatment 2 } & \text { Treatment 3 } \\45 & 30 & 39 \\41 & 34 & 35 \\37 & 35 & 38 \\40 & 40 &\end{array}$ At a 5% level of significance, test to see if there is a significant difference in the means of the three populations. (Please note that the sample sizes are not equal.) ​

In ANOVA, which of the following is NOT affected by whether or not the population means are equal?

An experimental design where the experimental units are randomly assigned to the treatments is known as _____.

In an analysis of variance where the total sample size for the experiment is nT and the number of populations is k, the mean square within treatments is _____.

In a completely randomized design involving four treatments, the following information is provided. $\begin{array}{l}&\text { Treatment 1 } & \text { Treatment 2 } & \text { Treatment 3 } & \text { Treatment 4 } \\\text { Sample size }&50 & 18 & 15 & 17 \\\text { Sample mean }&32 & 38 & 42 & 48\end{array}$ The overall mean (the grand mean) for all treatments is _____.

Three different models of automobiles (A, B, and C) were compared for gasoline consumption. For each model of car, fifteen cars were randomly selected and subjected to standard driving procedures. The average miles/gallon obtained for each model of car and sample standard deviations are shown below. $\begin{array}{l}&\text { Car } A & \text { Car B } & \text { Car C } \\\text { Average miles per gallon }&42 & 49 & 44 \\\text { Sample standard deviation }&4 & 5 & 3\end{array}$ Use the above data and test to see if the mean gasoline consumption for all three models of cars is the same. Let α = .05. ​

Individuals were randomly assigned to three different production processes. The hourly units of production for the three processes are shown below. $\begin{array}{l}\text { Production Process }\\\begin{array} { c c c } \text { Process 1 } & \text { Process 2 } & \text { Process 3 } \\33 & 33 & 28 \\30 & 35 & 36 \\28 & 30 & 30 \\29 & 38 & 34\end{array}\end{array}$ Use Excel with α = .05 to determine whether there is a significant difference in the mean hourly units of production for the three types of production processes.

Five drivers were selected to test drive two makes of automobiles. The following table shows the number of miles per gallon for each driver driving each car. $\begin{array}{llllll}&&&\text { Driver }\\\text { Automobile }&1&2&3&4&5\\A & 30 & 31 & 30 & 27 & 32 \\B & 36 & 35 & 28 & 31 & 30\end{array}$ Consider the makes of automobiles as treatments and the drivers as blocks and use Excel to determine whether there is any difference in the miles/gallon of the two makes of automobiles. Let α = .05.

An experimental design that permits statistical conclusions about two or more factors is a _____.

In a completely randomized design involving four treatments, the following information is provided. $\begin{array} { l } & \text {Treatment 1 }& \text {Treatment 2 }& \text {Treatment 3 }& \text {Treatment 4 }\\\hline \text {Sample size }&50&18&15&17\\\text { Sample mean }&32&38&42&48\\\end{array}$ The overall mean (the grand mean) for all treatments is _____.

The final examination grades of random samples of students from three different classes are shown below. $\begin{array} { c c c } \text { Class A } & \text { Class B } & \text { Class C } \\92 & 91 & 85 \\85 & 85 & 93 \\96 & 90 & 82 \\95 & 86 & 84\end{array}$ At the α = .05 level of significance, is there any difference in the mean grades of the three classes?

In an analysis of variance problem involving three treatments and 10 observations per treatment, SSE = 399.6. The MSE for this situation is _____.

In a completely randomized experimental design, 11 experimental units were used for each of the four treatments. Part of the ANOVA table is shown below. $\begin{array}{lllll}\text { Source of Variation } & \text { Sum of } & \text { Degrees of } & \text { Mean } & F \\& \text { Squares } & \text { Freedom } & \text { Square } &\\\hline \text {Between treatments }&1,500&\_\_\_\_?&\_\_\_\_?&\_\_\_\_?\\ \text { Within treatments (Error)}&\_\_\_\_?&\_\_\_\_?&\_\_\_\_?\\\hline \text { Total}&5,500\\\end{array}$ Fill in the blanks in the above ANOVA table. ​

In the ANOVA, treatment refers to _____.

For four populations, the population variances are assumed to be equal. Random samples from each population provide the following data. $\begin{array} { c c c c } \text { Population } & \text { Sample Size } & \text { Sample Mean } & \text { Sample Variance } \\1 & 11 & 40 & 23.4 \\2 & 11 & 35 & 21.6 \\3 & 11 & 39 & 25.2 \\4 & 11 & 37 & 24.6\end{array}$ Using a .05 level of significance, test to see if the means for all four populations are the same.

Three major automobile manufacturers have entered their cars in the Indianapolis 500 race. The speeds of the tested cars are given below. $\begin{array} { c c c } \text { Manufacturer A } & \text { Manufacturer B } & \text { Manufacturer C } \\\hline 180 & 177 & 175 \\175 & 180 & 176 \\179 & 167 & 177 \\176 & 172 & \\190 & &\end{array}$ At α = .05, use Excel to determine whether there is a significant difference in the average speeds of the cars of the auto manufacturers.