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An article in the ACI Materials Journal (Vol. 84, 1987, pp. 213–216) describes several experiments investigating the rodding of concrete to remove entrapped air. A 3-inch & 6-inch cylinder was used, and the number of times this rod was used is the design variable. The resulting compressive strength of the concrete specimen is the response. The data are shown in the following table:

Observations for Tensile Strength

(a) Is there any difference in compressive strength due to the rodding level? Use alpha= 0.05. (b) Find the P-value for the F statistic in part (a).
Rodding Level 1 2 3
10 1530 1530 1440
15 1610 1650 1500
20 1560 1730 1530
25 1500 1490 1510

Solution

rod_10<-c(1530,1530,1440)
rod_15<-c(1610,1650,1500)
rod_20<-c(1560,1730,1530)
rod_25<-c(1500,1490,1510)
rodadd<-rbind(rod_10,rod_15,rod_20,rod_25)
print(rodadd)
##        [,1] [,2] [,3]
## rod_10 1530 1530 1440
## rod_15 1610 1650 1500
## rod_20 1560 1730 1530
## rod_25 1500 1490 1510
r10<-mean(rod_10)
r15<-mean(rod_15)
r20<-mean(rod_20)
r25<-mean(rod_25)

Now determine the SSE, SStreatment, MSE, MStreatment

SSE_10 <- (1530-r10)^2 + (1530-r10)^2 + (1440-r10)^2
SSE_15 <- (1610-r15)^2 + (1650-r15)^2 + (1500-r15)^2
SSE_20 <- (1560-r20)^2 + (1730-r20)^2 + (1530-r20)^2
SSE_25 <- (1500-r25)^2 + (1490-r25)^2 + (1510-r25)^2

a) For SSE & MSE:

SSE<- SSE_10 + SSE_15 + SSE_20 + SSE_25 
print(SSE)
## [1] 40933.33
MSE<- SSE/(12-4)
print(MSE)
## [1] 5116.667

b) SStreatment & MStreatment:

mean <- c(mean(rodadd))
print(mean)
## [1] 1548.333
SStreatment <- 3*((r10 - mean)^2 + (r15 - mean)^2 + (r20 - mean)^2 + (r25 - mean)^2)
print(SStreatment)
## [1] 28633.33
MStreatment<-SStreatment/(4-1)
print(MStreatment)
## [1] 9544.444

c) Sum Squared Total (SST)

SST<-SSE+SStreatment
print(SST)
## [1] 69566.67

d) Finding F-Statistic:

Fo<-MStreatment/MSE
print(Fo)
## [1] 1.865364

e) Finding Critical Value of F: using F-Distribution

Fcritical<-qf(0.95,3,8)
print(Fcritical)
## [1] 4.066181

PART 2): Determine the P-Value

Pvalue<-pf(1.86536, 3, 8, lower.tail = FALSE)
print(Pvalue)
## [1] 0.2137821

Deduction

We have a F-statistics value of 1.865. This is below the critical value of 4.066. As a result, we do not reject the Ho.

Conclusion: There seems to be no difference between the rodding level and compressive strength.

P-value is 0.2138

Completer R Code

##Loading and Reading the data
rod_10<-c(1530,1530,1440)
rod_15<-c(1610,1650,1500)
rod_20<-c(1560,1730,1530)
rod_25<-c(1500,1490,1510)
rodadd<-rbind(rod_10,rod_15,rod_20,rod_25)
print(rodadd)


r10<-mean(rod_10)
r15<-mean(rod_15)
r20<-mean(rod_20)

r25<-mean(rod_25)

##Determine the SSE,SStreatment,MSE,MStreatment:
SSE_10 <- (1530-r10)^2 + (1530-r10)^2 + (1440-r10)^2
SSE_15 <- (1610-r15)^2 + (1650-r15)^2 + (1500-r15)^2
SSE_20 <- (1560-r20)^2 + (1730-r20)^2 + (1530-r20)^2
SSE_25 <- (1500-r25)^2 + (1490-r25)^2 + (1510-r25)^2


SSE<- SSE10 + SSE15 + SSE20 + SSE25
print(SSE)

#For MSE:
MSE<- SSE/(12-4)
print(MSE)

#For SStreatment::
mean <- c(mean(rodadd))
print(mean)

SStreatment <- 3*((r10 - mean)^2 + (r15 - mean)^2 + (r20 - mean)^2 + (r25 - mean)^2)
print(SStreatment)

#For MStreatment:
MStreatment<-SStreatment/(4-1)
print(MStreatment)

#For SST::
SST<-SSE+SStreatment
print(SST)

#For F-Statistic:
Fo<-MStreatment/MSE
print(Fo)

#For Critical Value:

Fcritical<-qf(0.95,3,8)
print(Fcritical)

#PART 2
Pvalue<-pf(1.86536, 3, 8, lower.tail = FALSE)
print(Pvalue)