Deck 11: One-Factor Anova: Fixed-Effects Mode

Question 1

A consumer testing lab wants to compare the mean life of AA batteries produced by different manufacturers. Five brands of batteries are selected, and for each brand, 20 batteries are randomly sampled. The lab then tests for the lifetime of each battery (in hours) and compares the average battery life of different brands using ANOVA. Complete the following one-factor ANOVA summary table using $\alpha$ = .05. Based on the results, do batteries of different brands have different lifetimes?

$\begin{array}{lccccc}\hline \text { Source } & S S & d f & M S & F & \text { Critical Value and Decision } \\\hline \text { B etween } & & & & & \\\text { Within } & & & 10 & & \\\text { Total } & 1110 & & & & \\\hline\end{array}$

Answer 1

There are 5 brands, so J = 5. Each brand has 20 batteries sampled, so n = 20. N = 5*20 = 100.

df_betw = J - 1 = 5 - 1 = 4, df_with = N - J = 100- 5 = 95, df_total = N - 1 = 100 - 1 = 99.

SS_with = MS_with*df_with = 10*95 = 950. SS_betw = SS_total - SS_with = 1100 - 950 = 160.

MS_betw = SS_betw/df_betw = 160/4 = 40.

F = MS_betw/MS_with = 40/10 = 4, critical value = _.05F_4,95 = 2.47.

Because F > critical F value, we reject H₀ and conclude that batteries of different brands have different lifetimes.

$\begin{array}{lrrrcc}\hline \text { Source } & S S & {d f} & M S & F & \text { Critical Value and Decision } \\\hline \text { B etween } & 160 & 4 & 40 & 4 & .05 F_{4,95}=2.47 \\\text { Within } & 950 & 95 & 10 & & \text { reject } H_{0} \\\text { Total } & 1110 & 99 & & & \\\hline\end{array}$

Question 2

A reading specialist would like to know whether the page layout has any consistent effect on children's reading speed. He printed the same story in three types of page layout (one-column, two-column, and three-column) and then randomly assigned 15 children to each group. The time each child took to finish reading is recorded and compared using the one-factor ANOVA model. Complete the following ANOVA summary table using $\alpha$ = .05. Based on the results, does page layout have an effect on the speed of reading?

$\begin{array}{lccccc}\hline \text { Source } & S S & d f & M S & F & \text { Critical Value and Decision } \\\hline \text { Between } & & & 9 & 1.8 & \\\text { Within } & & & & & \\\text { Total } & & & & & \\\hline\end{array}$

Answer 2

There are 3 different types of page layout, so J = 3. There were 15 children in each group, so n = 15.
N = 3*15 = 45.
df_betw = J - 1 = 3 - 1 = 2, df_with = N - J = 45 - 3 = 42, df_total = N - 1 = 45 - 1 = 44.
Because F = MS_betw/MS_with, MS_with = MS_betw/F = 9/1.8 = 5.

SS_betw = MS_betw*df_betw = 9*2 = 18, SS_with = MS_with*df_with = 5*42 = 210, SS_total = SS_betw + SS_with = 18 + 210 = 228.

Critical value _.05F_2,42 = 3.22. Because F = 1.8 < critical F value, we fail to reject H₀ and conclude that page layout does not have a significant effect on the speed of reading.

$\begin{array}{lrrrcc}\hline \text { Source } & S S & {\boldsymbol{d f}} & {M S} & F & \text { Critical Value and Decision } \\\hline \text { B etween } & 18 & 2 & 9 & 1.8 & .05 F_{2,42}=3.22 \\\text { Within } & 210 & 42 & 5 & & \text { Fail to reject } H 0 \\\text { Total } & 228 & 44 & & & \\\hline\end{array}$

Question 3

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.

$\begin{array}{ccc}\text { W-Mart } & \text { Tag } & \text { URToy } \\\hline 23 & 24 & 30 \\22 & 27 & 30 \\25 & 28 & 26 \\24 & 25 & 29 \\23 & 27 & 29 \\\hline\end{array}$
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.

Answer 3

Procedure:
1) Create a data set with two variables, Prices and Retailer. The data set should have 15 cases, each case representing one store.

2) 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 a 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.

"3) 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:
Tests of Between-Subjects Efects
Dependent Variable: Frices

$\begin{array}{ccccccc}\hline \text { Source } & \begin{array}{c}\text { Type III Sum of } \\\text { Squares }\end{array} & d f & \text { Mean Square } & F & \text { Sig. } & \begin{array}{c}\text { Partial Eta } \\\text { Squared }\end{array} \\\hline \text { Retailer } & 72.933 & 2 & 36.467 & 16.328 & .000 & .731 \\\text { Error } & 26.800 & 12 & 2.233 & & & \\\text { Corrected Total } & 99.733 & 14 & & & & \\\hline\end{array}$
Levene's Test of Equality of Error
Variances Dependent Variable: Prices

$\begin{array}{llll}\hline F & d f 2 & d f 2 & \text { Sig. } \\\hline 467 & 2 & 12 & .638 \\\hline\end{array}$
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 ( $\chi$ 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."

Question 4

A stock analyst wanted to compare the long-term return of stocks from different industries. She randomly selected eight stocks in each of the three industries of interest (financial, energy, utilities) and compiled the 10-year rate of return for each stock (assume the return for one stock is not dependent on the return for any other stock). Below are the data that were collected.

$\begin{array}{rrr}\text { Financial } & \text { Energy } & \text { Utilities } \\\hline 10.76 & 12.72 & 10.88 \\15.05 & 14.91 & 5.86 \\17.01 & 6.43 & 12.46 \\5.07 & 11.19 & 9.90 \\19.50 & 18.79 & 3.95 \\8.16 & 20.73 & 3.44 \\10.38 & 9.60 & 7.11 \\6.76 & 17.40 & 15.70 \\\hline\end{array}$
Use SPSS to conduct a one-factor ANOVA to determine if the returns are equal across industries ( $\alpha$ = .05).
Test the assumptions, plot the group means, consider an effect size, interpret the results, and write an APA-style summary.

Question 5

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 a one-factor ANOVA to analyze her data. The ANOVA table below summarized the results she obtained.

$\begin{array}{lccccc}\hline \text { Source } & S S & d f & M S & F & \text { Critical Value and Decision } \\\hline \text { B etween } & 600 & 3 & 200 & 0.5 & 05 F_{3,26}=2.975 \\\text { Within } & 2600 & 26 & 100 & & \text { Fail to reject } H 0 \\\text { Total } & 3200 & 30 & & & \\\hline\end{array}$

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?

Question 6

The ability of ANOVA to compare variation between groups is referring specifically to which one of the following?

A) Comparing variation between the categories of the independent variable
B) Comparing variation between the categories of the dependent variable
C) Comparing variation within cases within the same category of the independent variable
D) Comparing the relationship between the dependent variable and independent variable

Question 7

The ability of ANOVA to compare variation within groups is referring specifically to which one of the following?

A) Comparing variation between the categories of the independent variable
B) Comparing variation between the categories of the dependent variable
C) Comparing variation within cases within the same category of the independent variable
D) Comparing the relationship between the dependent variable and independent variable

Question 8

Which one of the following is appropriate for a one-way ANOVA?

A) One continuous dependent variable and one categorical independent variable with two or more groups
B) One categorical dependent variable and one categorical independent variable with two or more groups
C) Two or more continuous dependent variables and one categorical independent variable with two or more groups
D) One continuous dependent variable and two or more categorical independent variables with two or more groups