## Question

1.Brainstorming. A researcher investigated whether brainstorming is more effective for larger groups than for smaller ones by setting up four groups of agribusiness executives, the group sizes being two, three, four, and five, respectively. He also set up four groups of agribusiness scientists, the group sizes being the same as for the agribusiness executives. The researcher gave each group the same problem: “How can Canada increase the value of its agricultural exports?” Each group was allowed 30 minutes to generate ideas. The variable of interest was the number of different ideas proposed by the group. The results, classified by type of group (factor A) and size of group (factor B), were:

Table 1:

B1 (Two) B2 (Three) B3 (Four) B4 (Five)

A1 (Agribusiness executives) 18 22 31 32

A2 (Agribusiness scientists) 15 23 29 33

Assume that no-interaction ANOVA model is appropriate.

1. Plot the data in an interaction plot. Does it appear that interaction effects are present? Does it appear that factor A and factor B main effects are present? Discuss.

2. (By hand) Conduct separate tests for type of group and size of group main effects. In each test, use level of significance α = 0.01 and state the alternatives, decision rule, and conclusion. What is the P-value for each test?

3. (By hand) Obtain confidence intervals for D1 = μ·2 − μ·1 ,D2 = μ·3 − μ·2, D3 = μ·4 − μ·3; use the Bonferroni procedure with a 95 percent family confidence coefficient. State your findings.

## Solution Preview

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## 1```{r}

B = rep(c("B1", "B2", "B3", "B4"), 2)

A = c(rep("A1", 4), rep("A2",4))

value = c(18, 22, 31, 32, 15, 23, 29, 33)

df<-data.frame(A,B,value)

interaction.plot(x.factor = df$B, trace.factor = df$A,

response = df$value, type ="b", col = 2:3, xlab ="B", ylab ="Mean", trace.label ="B")

```

As the interaction lines are't parallel and crossing each other at three places. It looks

there is an interaction between two factors.

To test it anova model can be fitted:

```{r}

summary(aov(value ~ A + B, data=df))

```

From the results we conclude that only factor B (size of group) is significant and is the most significant factor. This suggests that changing group size would have significant impact

on values which is the main effect.

## 2

Given:

\begin{center}

\begin{tabular}{ |c|c|c|c|c| }

\hline

& B1 & B2 & B3 & B4 \\

\hline

A1 & 18 & 22 & 31 & 32 \\...

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