2.34 An article in the Journal of Strain Analysis (vol. 18, no. 2, 1983) compares several procedures for predicting the shear strength for steel plate girders. Data for nine girders in the form of the ratio of predicted to observed load for two of these procedures, the Karlsruhe and Lehigh methods, are as follows:

Entering sample data and Formation of data frame.

Karlsruhe_Method<-c(1.186,1.151,1.322,1.339,1.200,1.402,1.365,1.537,1.559)
Lehigh_Method<-c(1.061,0.992,1.063,1.062,1.065,1.178,1.037,1.086,1.052)

data<-data.frame(Karlsruhe_Method,Lehigh_Method)

Diff_bw<-c(Karlsruhe_Method-Lehigh_Method)

2.34(d) Investigate the normality assumption for both samples.

Normal Probility plot of Karlsruhe Method and Lehigh Method.

qqnorm(Karlsruhe_Method, main = "Normal Probability of Karlsruhe Method")
qqline(Karlsruhe_Method)

qqnorm(Lehigh_Method, main = "Normal Probability of Lehigh Method")
qqline(Lehigh_Method)

Comments: The plots of both Karlsruhe Method and Lehigh Method looks normally distributed, but plot of Karlsruhe Method seem left skewed, Lehigh Method has long Tails-out drifted. The plots could be assumed to be normally distributed.

2.34(e) Investigate the normality assumption for the difference in ratios for the two methods.

Normal Probability plot of difference between Karlsruhe Method and Lehigh Method.

qqnorm(Diff_bw, main = "Normal Probability plot of difference between Karlsruhe Method and Lehigh Method.")
qqline(Diff_bw)

Comments: The plot is left skewed and plot could be assumed to be normal because of the most of the points distributed lays along the line.

Box Plot of Karlsruhe Method and Lehigh Method.

boxplot(Karlsruhe_Method,Lehigh_Method, main = "Box plot of Karlsruhe Method and Lehigh Method", names = c("Karlsruhe Method", "Lehigh Method"))

Comments: The Karlsruhe Method has wide quartile range and samples distributed along the with max above 1.5 and min at 1.15 approx, but the Lehigh Method has no appearing quartile range and could only be seen as thick line. I don’t think we could pool the variance.

Test Hypothesis

\[ H_o:\mu1=\mu2 \\ H_a:\mu1\neq \mu2 \] Null Hypothesis(Ho), Alternative Hypothesis(Ha). where; μ1 = mean of Karlsruhe Method , μ2 = mean of Lehigh Method.

The Karlsruhe Method and Lehigh Method are related by the same Girder, so we use paired T-Test

T-test

t.test(Karlsruhe_Method,Lehigh_Method, alternative = "greater", paired = TRUE)
## 
##  Paired t-test
## 
## data:  Karlsruhe_Method and Lehigh_Method
## t = 6.0819, df = 8, p-value = 0.0001476
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
##  0.1901476       Inf
## sample estimates:
## mean of the differences 
##               0.2738889

2.34(a) Is there any evidence to support a claim that there is a difference in mean performance between the two methods? Use α = 0.05.

Answer: For the Paired t-test we could see that μ1 - μ2 = 0.27388, hence we could say there is difference in means.

2.34(b) What is the P-value for the test in part (a)?

Answer: P-value = 0.0001476.

2.34(c) Construct a 95 percent confidence interval for the difference in mean predicted to observed load.

Answer: 95 percent confidence interval: 0.1901476 inf.

2.34(f) Discuss the role of the normality assumption in the paired t-test.

Answer: The role of normality in paired t-test, due to the left skew in the normal plot of Karlsruhe Method, and long Tails-out drifted and short quartile range of Lehigh Method has given us the results of rejecting the Ho.

Source code:

# Entering sample data and data frame formation.

Karlsruhe_Method<-c(1.186,1.151,1.322,1.339,1.200,1.402,1.365,1.537,1.559)
Lehigh_Method<-c(1.061,0.992,1.063,1.062,1.065,1.178,1.037,1.086,1.052)

data<-data.frame(Karlsruhe_Method,Lehigh_Method)

Diff_bw<-c(Karlsruhe_Method-Lehigh_Method)


# Normal Probability plot of Karlsruhe Method and Lehigh Method.

qqnorm(Karlsruhe_Method, main = "Normal Probability of Karlsruhe Method")
qqline(Karlsruhe_Method)
qqnorm(Lehigh_Method, main = "Normal Probability of Lehigh Method")
qqline(Lehigh_Method)

# Normal Probability plot of difference between Karlsruhe Method and Lehigh Method.
qqnorm(Diff_bw, main = "Normal Probability plot of difference between Karlsruhe Method and Lehigh Method.")
qqline(Diff_bw)


# Box Plot
boxplot(Karlsruhe_Method,Lehigh_Method, main = "Box plot of Karlsruhe Method and Lehigh Method", names = c("Karlsruhe Method", "Lehigh Method"))


# T-test
t.test(Karlsruhe_Method,Lehigh_Method, alternative = "greater", paired = TRUE)