Question

Use the data from problem 7 to determine if there is interaction present in this study?

Cite specific statistics to support your claim.

How does the answer to part a effect the results of the study

Drying Time (min)

Paint 20 25 30

1 74 73 78

64 61 85

50 44 92

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2 92 98 66

86 73 45

68 88 85

Answer 1

Solution

The analysis is 2-Way ANOVA with equal number of observations per cell.

Let xijk be the kth value of the response variable under jth drying time and ith paint.

Then the ANOVA model is: xijk = µ + αi + βj + γij + εijk, where

µ = common effect,

αi = effect of ith row (paint),

βj = effect of jth column (drying time), γij = row-column (paint-drying time) interaction and εijk is the error component which is assumed to be Normally Distributed with mean 0 and variance σ².

Hypotheses:

Null hypothesis: H₀₁: α₁ = α₂   Vs Alternative: H₁₁: α₁ and α₂ are different.

Null hypothesis: H₀₂: β₁ = β₂ = β₃ Vs Alternative: H₁₂: at least one βj is different from other βj’s.    

Null hypothesis: H₀₃: γij = 0 for all i and j   Vs Alternative: H₁₃: at least one γij is not zero.

Now to work out the solution,

ANOVA TABLE [α = 0.05]
Source	DF	SS		MSS		Fobs		Fcrit		p-value
Row	1	355.5556		355.5556		1.902497		4.747225		0.192963
Column	2	27.44444		13.72222		0.073424		3.885294		0.92962
Interaction	2	1878.778		939.3889		5.026457		3.885294		0.025959
Between	5	2261.778		452.3556
Error	12	2242.667		186.8889
Total	17	4504.444		264.9673

Since Fobs > Fcrit, for Interaction, H₀₃is rejected. Hence, we conclude that there is enough evidence to suggest the interaction between paint and drying time does impact the response variable. ANSWER