Use the data in http://personal.psu.edu/acq/401/Data/PorousCarbon.txt to compare

Question

Use the data in http://personal.psu.edu/acq/401/Data/PorousCarbon.txt to compare the mean pore size of carbon made at four different temperatures (Problem 6, Section 10.2, p.359):

This link will take you to Problem 6, Section 10.2, p.359) https://www.chegg.com/homework-help/Probability-amp-Statistics-with-R-for-Engineers-and-Scientists-1st-edition-chapter-10.2-problem-6E-solution-9780321852991

a.) Test the assumptions of homoscedasticity and normality. Report the p-values and return the corresponding plots. Are the plots in agreement with the p-values?

b.) Conduct the F-test. Copy and paste the R output, and give the p-value for the F test. Is H0 rejected at level 0.05?

[NOTE: You cannot just use temp instead of ind in the commands given in Example 1, because it is numerical (R will fit the simple linear regression model). Right after you read the data into, say pc, issue the command pc$ind=as.factor(pc$temp). Then you can use ind as in Example 1.]

c.) Perform Bonferroni’s MCs and SCIs either through the T-test, or the rank-sum test. Also, perform Tukey’s MCs and SCIs at experiment-wise level of significance = 0.05. Copy and paste the commands, and state the conclusion of the multiple comparisons quoting either p-values or the SCIs.

Thank you!

Explanation / Answer

a) Test the assumptions of homoscedasticity and normality b) F-test

v=c(7.75,7,6.9,7.5,8,6.9,7.6,7,7,7.7,6.7,6.9,6.9,6.2,6.6,6.4,6.2,6,6.6,6)
temp=c(300,300,300,300,300,400,400,400,400,400,500,500,500,500,500,600,600,600,600,600)
res.aov <- aov(v~temp) # Compute the analysis of variance
summary(res.aov) # Summary of the analysis
oneway.test(v~temp) # one-way test to check the homogeneity
par(mfrow=c(1,2))
plot(res.aov, 1)
plot(res.aov, 2)
aov_residuals <- residuals(object = res.aov ) # Extract the residuals
shapiro.test(x = aov_residuals ) # Run Shapiro-Wilk test

> v=c(7.75,7,6.9,7.5,8,6.9,7.6,7,7,7.7,6.7,6.9,6.9,6.2,6.6,6.4,6.2,6,6.6,6)
> temp=c(300,300,300,300,300,400,400,400,400,400,500,500,500,500,500,600,600,600,600,600)
> res.aov <- aov(v~temp) # Compute the analysis of variance
> summary(res.aov) # Summary of the analysis
            Df Sum Sq Mean Sq F value   Pr(>F)  
temp         1 4.306   4.306   34.98 1.34e-05 ***
Residuals   18 2.216   0.123                   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> oneway.test(v~temp) # one-way test to check the homogeneity

        One-way analysis of means (not assuming equal variances)

data: v and temp
F = 10.903, num df = 3.0000, denom df = 8.7155, p-value = 0.002606

> par(mfrow=c(1,2))
> plot(res.aov, 1)
> plot(res.aov, 2)

As all the points fall approximately along this reference line, we can assume normality.
> aov_residuals <- residuals(object = res.aov ) # Extract the residuals
> shapiro.test(x = aov_residuals ) # Run Shapiro-Wilk test

        Shapiro-Wilk normality test

data: aov_residuals
W = 0.97345, p-value = 0.8255

The p-balue is less than level of significance, we reject the null hypothesis

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