Question

A study was conducted to determine the effects of two factors on the latherability of soap. The two factors were type of water (tap or distilled) and glycerol (present or absent). The outcome measured was the amount of foam produced (in milliliters). The experiment was repeated three times for each combination of the factors. The results are as follows.

	No Glycerol	Glycerol
Distilled Water	165   181   168	170   197   190
Tap Water	155   142   139	139   160   160

Analyze the data using a=.05.

Answer 1

Solution:

Given that

A study was conducted to determine the effects of two factors on the latherability of soap. The two factors were type of water (tap or distilled) and glycerol (present or absent).

Let A denote the factor type of water with two levels A₁,A₂ and B denote the factor presence of glycerol with two levels B₁,B₂ .

Let X_ijk denote the amount of foam produced by the k th soap tested of type (A_i,B_j) where i=1,2,j=1,2,k=1,2,3.

The appropriate linear model is given by

   i,j = 1,2 ,k=1,2,3

where = general effect due to both the factors

= additional effect due to ith factor of A

= additional effect due to jth factor of B

= interaction effect due to ith level of A and jth level of B

= error term associated

Assume ~ N(0,) i.i.d. and , , , .

To test     ag. H₁: Not H₀.

The 2 way anova table is given by

Df Sum Sq Mean Sq F value Pr(>F)   
a 1 2581.3 2581.3 21.158 0.00176
b 1 363.0 363.0 2.975 0.12282   
a:b 1 33.3 33.3 0.273 0.61534   
error 8 976.0 122.0   

(The R code to obtain this table is

x=c(165,181,168,170,197,190,155,142,139,139,160,160)
a=gl(2,6)
b=as.factor(c(1,1,1,2,2,2,1,1,1,2,2,2))
fit=aov(lm(x~a+b+a*b))
summary(fit)

)

Since p-value(AB) = 0.61534 > =0.05 we accept H_0AB and conclude that at 5% level of significance no significant interaction is present.

Now we are interested to test the differential effects i.e. and .

From the anova table we find

p-value(A) = 0.00176 < =0.05 ------- We reject H_0A and conclude at 5% level of significance that type of water has differential effect on amount of foam.

p-value(B) = 0.12282 > =0.05 -------- We accept H_0B and conclude at 5% level of significance that presence of glycerol has no significant effect on amount of foam.