An experiment was conducted to determine if either firing temperature or furnace position affects the baked density of a carbon anode. The data are shown below.

1. Write your experimental question. Use α=0.05.

Is there a significant effect of firing temperature, furnace position and the interaction of firing temperature and furnace position in the baked density of a carbon node at α=0.05?

2. Construct your null and alternative hypotheses.

The hypothesis being tested are the following:

For firing temperature:

Ho: There is no significant main effect of firing temperature in the baked density of a carbon anode. Ha: There is a significant main effect of firing temperature in the baked density of a carbon anode.

For furnace position:

Ho: There is no significant main effect of furnace position in the baked density of a carbon anode. Ha: There is a significant main effect of furnace position in the baked density of a carbon anode.

For firing temperature and furnace position interaction:

Ho: There is no significant interaction effect between firing temperature and furnace position in the baked density of a carbon anode. Ha: There is a significant interaction effect between firing temperature and furnace position in the baked density of a carbon anode.

3. Suppose we assume that no interaction exists.

(a) Write down the statistical model.

The model for the two-factor, no interaction model is \(Yijk=μ+τi+βi+ϵijk\) where, \(μ\) is the overall mean of all the runs, \(τi\) is the effect of furnace position, \(βi\) is the effect of firing temperature and \(ϵijk\) is the random error.

(b) Conduct the analysis of variance assuming both treatments to be fixed effects.

(c) What conclusions/interpretations can be drawn?

From the table above, we can see that th p-value is greater than α=0.05 and we have \(F_{0.05,2,12}=3.89\). We can observe that Firing temperature has an F-value equal to 1056.117 which is clearly greater than \(F_{0.05,2,1,2}=3.89\). Hence, we reject the null hypothesis. Additionally, the F-value of Furnace position is 15.998 which is also greater than \(F_{0.05,2,1,2}=3.89\). Thus, we reject the null. Also, Furnace and Firing temperature has an F-value of 0.914 which is less than \(F_{0.05,2,1,2}=3.89\). Hence, we failed to reject the null hypothesis.

(d)Derive the expected mean squares.

The expected mean squares (EMS) reveal that firing temperature exerts the greatest influence on the observed outcome, with an EMS of 472671 dominating that of furnace position (7160). This highlights the significantly larger variability caused by temperature compared to position. Additionally, the relatively small error term (448) indicates minimal random error.

(e) Comment on model adequacy.

4. Perform a post-hoc test using Tukey multiple comparisons of means. Use α=0.05. Provide interpretations of your findings.

The data shows a statistically significant difference of -39.88889 between the two furnace positions with its p-value less than 0.05.

The data shows that all two pairs of means differ except the pair of temperatures of 850 and 800.

5. Give details on the syntax used to produce your answer.

This code is use to export my data from excel.

library(readxl)
LAB4<-read_xlsx("D:/stat//LAB4.xlsx")

Used this code to show the given data

library(knitr)
library(kableExtra)

kable(LAB4, format = "html") %>%
  kable_styling(full_width = FALSE) %>%
  row_spec(0, bold = TRUE, color = "black", background = "lightgray") %>%
  row_spec(1:4, background = "white")
POSITION 800°C 825°C 850°C
1 570 1063 565
1 565 1080 510
1 583 1043 590
2 528 988 526
2 547 1026 538
2 521 1004 532

This code is for the ANOVA table.

library(dplyr)
runs<- c(570,1063,565,565,1080,510,583,1043,590,528,988,526,547,1026,538,521,1004,532)
temp<- c(800,825,850)
pos<-c(rep(1,9),rep(2,9))
temp<-c(rep(temp,6))
library(GAD)
pos <- as.fixed(pos)
temp <- as.fixed(temp)
model <- aov(runs~pos+temp+pos*temp)
summary(model)
            Df Sum Sq Mean Sq  F value   Pr(>F)    
pos          1   7160    7160   15.998  0.00176 ** 
temp         2 945342  472671 1056.117 3.25e-14 ***
pos:temp     2    818     409    0.914  0.42711    
Residuals   12   5371     448                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

To plot my data for analysis

plot(model)

I use TukeyHSD() and store in the variable tukey_result to compute for post-hoc test.

tukey_result <- TukeyHSD(model, "pos")
tukey_result
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = runs ~ pos + temp + pos * temp)

$pos
         diff       lwr       upr     p adj
2-1 -39.88889 -61.61776 -18.16002 0.0017624
tukey_result <- TukeyHSD(model, "temp")
tukey_result
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = runs ~ pos + temp + pos * temp)

$temp
               diff        lwr        upr     p adj
825-800  481.666667  449.08101  514.25232 0.0000000
850-800   -8.833333  -41.41899   23.75232 0.7547952
850-825 -490.500000 -523.08566 -457.91434 0.0000000