Suppose n1 = 15, n2 = 13, n3 = 17, n4 =17. For a one-factor ANOVA, the dfwith would be

A) 3 B) 4 C) 58 D) 61 A) 3 B) 4 C) 58 D) 61

Exam 11: One-Factor Anova: Fixed-Effects Model

A consumer group wanted to determine if there was a difference in prices for a specific type of toy depending on where the toy was purchased. In the local area there are three main retailers: W-Mart, Tag, and URToy. For each retailer, the consumer group randomly selected five stores located in different parts of the city, and collected their listed prices of that specific type of toy (in dollars). Assume that all stores priced their merchandise independently. W-Mart Tag URToy 23 24 30 22 27 30 25 28 26 24 25 29 23 27 29 Use SPSS to conduct a one-factor ANOVA to determine if the prices are different across different retailers using $\alpha$ = .05. Test the assumptions, plot the group means, consider an effect size, interpret the results, and write an APA-style summary.

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Question 1

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Procedure:
Create a data set with two variables, Prices and Retailer. The data set should have 15 cases, each case representing one store.
Go to Analyze $\rightarrow$ General Linear Model $\rightarrow$ Univariate. Select Prices as the Dependent Variable and Retailer as the Fixed Factor. Go to Plot and select Retailer to the Horizontal Axis, then Add, to get profile plot. Go to Save and check Unstandardized under Residuals to save model residuals. Go to Options. Check Estimates of effect size to get effect size estimates. Check Homogeneity tests to examine the assumption of homoscedasticity.
To examine the assumption of independence, go to Graphs $\rightarrow$ Legacy Dialogs $\rightarrow$ Scatter/Dot $\rightarrow$ Simple Scatter $\rightarrow$ Define. Select RES_1 as the Y Axis, and Retailer as the X Axis. To examine the assumption of normality, go to Analyze $\rightarrow$ Descriptive Statistics $\rightarrow$ Explore. Select RES_1 to Dependent List. Go to Plots, and check Normality plots with tests.
Selected SPSS output:
$\text { Tests of Between-Subjects Effects }$
$\text { Dependent Variable: Frices }$
$\begin{array}{ccccccc}\hline \text { Source } & \begin{array}{c}\text { Type III Sum of } \\\text { Squares }\end{array} & \text { df } & \text { Mean Square } & \text { F } & \text { Sig. } & \begin{array}{c}\text { Partial Eta } \\\text { Squared }\end{array} \\\hline \text { Retailer } & 72.933 & 2 & 36.467 & 16.328 & .000 & .731 \\\text { Error } & 26.800 & 12 & 2.233 & & & \\\text { Corrected Total } & 99.733 & 14 & & & & \\\hline\end{array}$ $\text { Levene's Test of Equality of Error Variances }$
$\text { Dependent } V \text { ariable: Prices }$
$\begin{array}{llll}\hline \text { F } & \text { dfl } & \text { df2 } & \text { Sig. } \\\hline .467 & 2 & 12 & .638\\\hline \end{array}$ Profile plot
$Procedure: Create a data set with two variables, Prices and Retailer. The data set should have 15 cases, each case representing one store. Go to Analyze \rightarrow General Linear Model \rightarrow Univariate. Select Prices as the Dependent Variable and Retailer as the Fixed Factor. Go to Plot and select Retailer to the Horizontal Axis, then Add, to get profile plot. Go to Save and check Unstandardized under Residuals to save model residuals. Go to Options. Check Estimates of effect size to get effect size estimates. Check Homogeneity tests to examine the assumption of homoscedasticity. To examine the assumption of independence, go to Graphs \rightarrow Legacy Dialogs \rightarrow Scatter/Dot \rightarrow Simple Scatter \rightarrow Define. Select RES_1 as the Y Axis, and Retailer as the X Axis. To examine the assumption of normality, go to Analyze \rightarrow Descriptive Statistics \rightarrow Explore. Select RES_1 to Dependent List. Go to Plots, and check Normality plots with tests. Selected SPSS output: \text { Tests of Between-Subjects Effects } \text { Dependent Variable: Frices } \begin{array}{ccccccc} \hline \text { Source } & \begin{array}{c} \text { Type III Sum of } \\ \text { Squares } \end{array} & \text { df } & \text { Mean Square } & \text { F } & \text { Sig. } & \begin{array}{c} \text { Partial Eta } \\ \text { Squared } \end{array} \\ \hline \text { Retailer } & 72.933 & 2 & 36.467 & 16.328 & .000 & .731 \\ \text { Error } & 26.800 & 12 & 2.233 & & & \\ \text { Corrected Total } & 99.733 & 14 & & & & \\ \hline \end{array} \text { Levene's Test of Equality of Error Variances } \text { Dependent } V \text { ariable: Prices } \begin{array}{llll} \hline \text { F } & \text { dfl } & \text { df2 } & \text { Sig. } \\ \hline .467 & 2 & 12 & .638\\ \hline \end{array} Profile plot Residual plot by group Q-Q plot of residuals A one-way ANOVA was conducted to determine if the prices of a certain type of toy differed in three major retail stores. The Q-Q plot of residuals showed that the points clustered close to the diagonal line, suggesting that the assumption of normality was reasonable. According to Levene's test, the homogeneity of variance assumption was satisfied [F(2, 12) = .467, p = .638]. The scatterplot of residuals against the levels of the independent variable demonstrated a random display of points around 0, providing evidence to the assumption of independence being satisfied. From the ANOVA summary table, we see that the prices are significantly different across the three retailers (F = 16.328, df = 2,12, p <.001), the effect size is rather large ( \eta 2 = .731; suggesting about 73.1% of the variance in prices is accounted for by the differences in retailers). The profile plot suggested that the price is the lowest in W-Mart, higher in Tag, and the highest in URToy.$ Residual plot by group
$Procedure: Create a data set with two variables, Prices and Retailer. The data set should have 15 cases, each case representing one store. Go to Analyze \rightarrow General Linear Model \rightarrow Univariate. Select Prices as the Dependent Variable and Retailer as the Fixed Factor. Go to Plot and select Retailer to the Horizontal Axis, then Add, to get profile plot. Go to Save and check Unstandardized under Residuals to save model residuals. Go to Options. Check Estimates of effect size to get effect size estimates. Check Homogeneity tests to examine the assumption of homoscedasticity. To examine the assumption of independence, go to Graphs \rightarrow Legacy Dialogs \rightarrow Scatter/Dot \rightarrow Simple Scatter \rightarrow Define. Select RES_1 as the Y Axis, and Retailer as the X Axis. To examine the assumption of normality, go to Analyze \rightarrow Descriptive Statistics \rightarrow Explore. Select RES_1 to Dependent List. Go to Plots, and check Normality plots with tests. Selected SPSS output: \text { Tests of Between-Subjects Effects } \text { Dependent Variable: Frices } \begin{array}{ccccccc} \hline \text { Source } & \begin{array}{c} \text { Type III Sum of } \\ \text { Squares } \end{array} & \text { df } & \text { Mean Square } & \text { F } & \text { Sig. } & \begin{array}{c} \text { Partial Eta } \\ \text { Squared } \end{array} \\ \hline \text { Retailer } & 72.933 & 2 & 36.467 & 16.328 & .000 & .731 \\ \text { Error } & 26.800 & 12 & 2.233 & & & \\ \text { Corrected Total } & 99.733 & 14 & & & & \\ \hline \end{array} \text { Levene's Test of Equality of Error Variances } \text { Dependent } V \text { ariable: Prices } \begin{array}{llll} \hline \text { F } & \text { dfl } & \text { df2 } & \text { Sig. } \\ \hline .467 & 2 & 12 & .638\\ \hline \end{array} Profile plot Residual plot by group Q-Q plot of residuals A one-way ANOVA was conducted to determine if the prices of a certain type of toy differed in three major retail stores. The Q-Q plot of residuals showed that the points clustered close to the diagonal line, suggesting that the assumption of normality was reasonable. According to Levene's test, the homogeneity of variance assumption was satisfied [F(2, 12) = .467, p = .638]. The scatterplot of residuals against the levels of the independent variable demonstrated a random display of points around 0, providing evidence to the assumption of independence being satisfied. From the ANOVA summary table, we see that the prices are significantly different across the three retailers (F = 16.328, df = 2,12, p <.001), the effect size is rather large ( \eta 2 = .731; suggesting about 73.1% of the variance in prices is accounted for by the differences in retailers). The profile plot suggested that the price is the lowest in W-Mart, higher in Tag, and the highest in URToy.$ Q-Q plot of residuals
$Procedure: Create a data set with two variables, Prices and Retailer. The data set should have 15 cases, each case representing one store. Go to Analyze \rightarrow General Linear Model \rightarrow Univariate. Select Prices as the Dependent Variable and Retailer as the Fixed Factor. Go to Plot and select Retailer to the Horizontal Axis, then Add, to get profile plot. Go to Save and check Unstandardized under Residuals to save model residuals. Go to Options. Check Estimates of effect size to get effect size estimates. Check Homogeneity tests to examine the assumption of homoscedasticity. To examine the assumption of independence, go to Graphs \rightarrow Legacy Dialogs \rightarrow Scatter/Dot \rightarrow Simple Scatter \rightarrow Define. Select RES_1 as the Y Axis, and Retailer as the X Axis. To examine the assumption of normality, go to Analyze \rightarrow Descriptive Statistics \rightarrow Explore. Select RES_1 to Dependent List. Go to Plots, and check Normality plots with tests. Selected SPSS output: \text { Tests of Between-Subjects Effects } \text { Dependent Variable: Frices } \begin{array}{ccccccc} \hline \text { Source } & \begin{array}{c} \text { Type III Sum of } \\ \text { Squares } \end{array} & \text { df } & \text { Mean Square } & \text { F } & \text { Sig. } & \begin{array}{c} \text { Partial Eta } \\ \text { Squared } \end{array} \\ \hline \text { Retailer } & 72.933 & 2 & 36.467 & 16.328 & .000 & .731 \\ \text { Error } & 26.800 & 12 & 2.233 & & & \\ \text { Corrected Total } & 99.733 & 14 & & & & \\ \hline \end{array} \text { Levene's Test of Equality of Error Variances } \text { Dependent } V \text { ariable: Prices } \begin{array}{llll} \hline \text { F } & \text { dfl } & \text { df2 } & \text { Sig. } \\ \hline .467 & 2 & 12 & .638\\ \hline \end{array} Profile plot Residual plot by group Q-Q plot of residuals A one-way ANOVA was conducted to determine if the prices of a certain type of toy differed in three major retail stores. The Q-Q plot of residuals showed that the points clustered close to the diagonal line, suggesting that the assumption of normality was reasonable. According to Levene's test, the homogeneity of variance assumption was satisfied [F(2, 12) = .467, p = .638]. The scatterplot of residuals against the levels of the independent variable demonstrated a random display of points around 0, providing evidence to the assumption of independence being satisfied. From the ANOVA summary table, we see that the prices are significantly different across the three retailers (F = 16.328, df = 2,12, p <.001), the effect size is rather large ( \eta 2 = .731; suggesting about 73.1% of the variance in prices is accounted for by the differences in retailers). The profile plot suggested that the price is the lowest in W-Mart, higher in Tag, and the highest in URToy.$ A one-way ANOVA was conducted to determine if the prices of a certain type of toy differed in three major retail stores. The Q-Q plot of residuals showed that the points clustered close to the diagonal line, suggesting that the assumption of normality was reasonable. According to Levene's test, the homogeneity of variance assumption was satisfied [F(2, 12) = .467, p = .638]. The scatterplot of residuals against the levels of the independent variable demonstrated a random display of points around 0, providing evidence to the assumption of independence being satisfied. From the ANOVA summary table, we see that the prices are significantly different across the three retailers (F = 16.328, df = 2,12, p <.001), the effect size is rather large ( $\eta$ ² = .731; suggesting about 73.1% of the variance in prices is accounted for by the differences in retailers). The profile plot suggested that the price is the lowest in W-Mart, higher in Tag, and the highest in URToy.

A researcher was interested in comparing rental rates in four different parts of the city. She randomly selected a block from each part of the city. For each block, she collected the rental rates of different neighboring apartments. She then used one-factor ANOVA to analyze her data. The ANOVA table below summarized the results she obtained. Source SS df MS F Critical Value and Decision Between 600 3 200 0.5 .05=2.975 Within 2600 26 100 Fail to reject Total 3200 30 a. There are two mistakes in the ANOVA table. Identify the mistakes and correct them. b. Based on the research design, do you think any assumption of ANOVA may have been violated in this study? If so, what assumption is being violated? What might be the consequences of the violation?

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Question 2

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a. df_total = N - 1 = 29. F = MS_betw/MS_with = 2.
b. The assumption of independence may have been violated. The rental rates of neighboring apartments in the same block are possibly related with one another. Violation of this assumption would lead to increased likelihood of a Type I and/or Type II error in the F statistic; it also influences standard errors of means and thus inferences about those means.

Suppose that for a one-factor ANOVA with J = 3 and n = 10, the three sample means are all equal to 4.5. What is the value of MS_betw?

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Suppose that for a one-factor ANOVA with J = 3 and n = 10, the three sample means are all equal to 4.5. What is the value of MS_betw?

For a one-factor ANOVA comparing five groups with n = 30 in each group, the F ratio has degrees of freedom equal to

In ANOVA, which of the following is used to determine the appropriate F ratio?

When analyzing mean differences between three samples, doing all pairwise independent t tests instead of ANOVA using the same $\alpha$ level

In ANOVA, which of the following statements is possible?

In ANOVA, the variability between group means is estimated by

For J = 2 and $\alpha$ = .05, if the result of the one-factor fixed-effects ANOVA is nonsignificant, then the result of the independent t test using the same data set and same $\alpha$ level will be

In ANOVA, the average deviation of all the scores from their respective group means is estimated by

If you find an F ratio of 0.5 in a one-factor ANOVA ( $\alpha$ = .05), you will

Which of the following statements about $\alpha$ level is true?

Which of the following is a necessary assumption of the ANOVA model?

Using the same data with SPSS, Jamie and John found out that the p value of an ANOVA F test is 0.004. They both rejected the null hypothesis. However, Jamie used $\alpha$ = 0.05, whereas John used $\alpha$ = 0.01. Who has a higher probability of actually making a Type I error?

When analyzing mean differences between two samples, doing an independent t test instead of an ANOVA using the same  level

Suppose n₁ = 15, n₂ = 13, n₃ = 17, n₄ =17. For a one-factor ANOVA, the df_with would be

Introduction

Data Representation

Univariate Population Parameters and Sample Statistics

Normal Distribution and Standard Scores

Introduction to Probability and Sample Statistics

Inferences About a Single Mean

Inferences About the Difference Between Two Means

Inferences About Proportions

Inferences About Variances

Bivariate Measures of Association

Multiple Comparison Procedures

Factorial Anova: Fixed-Effects Model

One-Factor Fixed-Effects Ancova With Single Covariate

Random- and Mixed-Effects Analysis of Variance Models

Hierarchical and Randomized Block Analysis of Variance Models

Simple Linear Regression

Multiple Regression

Logistic Regression

Filters

Exam 11: One-Factor Anova: Fixed-Effects Model

Suppose that for a one-factor ANOVA with J = 3 and n = 10, the three sample means are all equal to 4.5. What is the value of MSbetw?

For a one-factor ANOVA comparing five groups with n = 30 in each group, the F ratio has degrees of freedom equal to

In ANOVA, which of the following is used to determine the appropriate F ratio?

When analyzing mean differences between three samples, doing all pairwise independent t tests instead of ANOVA using the same α\alphaα level

In ANOVA, which of the following statements is possible?

In ANOVA, the variability between group means is estimated by

For J = 2 and α\alphaα = .05, if the result of the one-factor fixed-effects ANOVA is nonsignificant, then the result of the independent t test using the same data set and same α\alphaα level will be

In ANOVA, the average deviation of all the scores from their respective group means is estimated by

If you find an F ratio of 0.5 in a one-factor ANOVA ( α\alphaα = .05), you will

Which of the following statements about α\alphaα level is true?

Which of the following is a necessary assumption of the ANOVA model?

Using the same data with SPSS, Jamie and John found out that the p value of an ANOVA F test is 0.004. They both rejected the null hypothesis. However, Jamie used α\alphaα = 0.05, whereas John used α\alphaα = 0.01. Who has a higher probability of actually making a Type I error?

When analyzing mean differences between two samples, doing an independent t test instead of an ANOVA using the same  level

Suppose n1 = 15, n2 = 13, n3 = 17, n4 =17. For a one-factor ANOVA, the dfwith would be

Introduction

Data Representation

Univariate Population Parameters and Sample Statistics

Normal Distribution and Standard Scores

Introduction to Probability and Sample Statistics

Inferences About a Single Mean

Inferences About the Difference Between Two Means

Inferences About Proportions

Inferences About Variances

Bivariate Measures of Association

Multiple Comparison Procedures

Factorial Anova: Fixed-Effects Model

One-Factor Fixed-Effects Ancova With Single Covariate

Random- and Mixed-Effects Analysis of Variance Models

Hierarchical and Randomized Block Analysis of Variance Models

Simple Linear Regression

Multiple Regression

Logistic Regression

Filters

Suppose that for a one-factor ANOVA with J = 3 and n = 10, the three sample means are all equal to 4.5. What is the value of MS_betw?

When analyzing mean differences between three samples, doing all pairwise independent t tests instead of ANOVA using the same $\alpha$ level

For J = 2 and $\alpha$ = .05, if the result of the one-factor fixed-effects ANOVA is nonsignificant, then the result of the independent t test using the same data set and same $\alpha$ level will be

If you find an F ratio of 0.5 in a one-factor ANOVA ( $\alpha$ = .05), you will

Which of the following statements about $\alpha$ level is true?

Using the same data with SPSS, Jamie and John found out that the p value of an ANOVA F test is 0.004. They both rejected the null hypothesis. However, Jamie used $\alpha$ = 0.05, whereas John used $\alpha$ = 0.01. Who has a higher probability of actually making a Type I error?

Suppose n₁ = 15, n₂ = 13, n₃ = 17, n₄ =17. For a one-factor ANOVA, the df_with would be