Ryan is a chemical engineer who is interested in comparing t

Ryan is a chemical engineer who is interested in comparing the hardness of four blends of paint. He applies the different paint blends to multiple pieces of metal, then cures the pieces. He takes six random samples from each paint blend, and measures each sample for hardness. He runs an ANOVA on the data, but spills paint on the output. This results in an incomplete table, as shown on the right. Complete the missing entries for the ANOVA table.

Using the ANOVA output, Ryan decides that at least one of the paint blends have a significantly different hardness. He then runs a Tukey test and obtains the output on the right. Which blend(s) were significantly different at the = 0.05 level of significance? Explain your reasoning.

Suppose Ryan conducts a completely different experiment to test whether the type of metal as well as the type of paint blend affects hardness. This time, he runs a two-way ANOVA analysis instead of one-way ANOVA analysis. He examines the interaction plot to the right. Is there evidence to suggest a significant interaction effect? Explain your reasoning.

Solution

Number of treatments (t) = number of blends of paint = 4

sample size from each paint blend = 6

The hypothesis for the test is,

H0 : All the paint blends have a same hardness.(mu1 = mu2 = mu3 = mu4)

H1 : at least one of the paint blends have a significantly different hardness.

First we find degrees of freedoms :

degrees of freedom for treatments :

t - 1 = 4 - 1 = 3

Degrees of freedom for total :

N - 1 = (6*4)-1 = 23

Degrees of freedom for error :

23 - 3 = 20

Sum of squares for treatments :

SST = SSTot - SSE = 593.766 - 312.068 = 281.698

Now calculate Mean square for treatments.

MST = SST / df

= 281.698 / 3 = 93.8993

F = MST / MSE = 93.8993 / 15.603 = 6.0180

P-value = 0.0043

alpha = 0.05

P-value < alpha

Reject H0 at 5% level of significance.

Conclusion : at least one of the paint blends have a significantly different hardness.

The complete ANOVA table will be,

The interaction plot indicate that there is interaction between hardness and paint.

The  two-way ANOVA results indicate that the interaction between hardness and paint is significant.