knitr::include_graphics("C:/Users/ADMIN/Downloads/UMP SEM 5/EDA/pdf2png/lab 3/lab 3-1.png")
knitr::include_graphics("C:/Users/ADMIN/Downloads/UMP SEM 5/EDA/pdf2png/lab 3/lab 3-2.png")
knitr::include_graphics("C:/Users/ADMIN/Downloads/UMP SEM 5/EDA/pdf2png/lab 3/lab 3-3.png")
knitr::include_graphics("C:/Users/ADMIN/Downloads/UMP SEM 5/EDA/pdf2png/lab 3/lab 3-4.png")
knitr::include_graphics("C:/Users/ADMIN/Downloads/UMP SEM 5/EDA/pdf2png/lab 3/lab 3-5.png")
Question 1
i. Two conditions of balanced incomplete block design
a) All pairs of treatments occur together within a block an equal number of times.
b) Any treatment p-value < α = 0.05 is significant in treatment comparison.
ii. Response variable : The strength of the paper produced
iii. Treatment factor : Hardwood concentrations
iv. List treatments: 2,4,6,8,10,12,14
library(readr)
## Warning: package 'readr' was built under R version 4.0.5
datalab3<- read_csv("C:/Users/ADMIN/Downloads/UMP SEM 5/EDA/datalabreport3.csv")
## Rows: 21 Columns: 3
## -- Column specification --------------------------------------------------------
## Delimiter: ","
## dbl (3): Hardwood, Days, Strength
##
## i Use `spec()` to retrieve the full column specification for this data.
## i Specify the column types or set `show_col_types = FALSE` to quiet this message.
datalab3
## # A tibble: 21 x 3
## Hardwood Days Strength
## <dbl> <dbl> <dbl>
## 1 2 1 114
## 2 2 5 120
## 3 2 7 117
## 4 4 1 126
## 5 4 2 120
## 6 4 6 119
## 7 6 2 137
## 8 6 3 117
## 9 6 7 134
## 10 8 1 141
## # ... with 11 more rows
Treatment = as.factor(datalab3$Hardwood)
block = as.factor(datalab3$Days)
Question 4
results = aov(Strength~block+Treatment,datalab3)
summary(results)
## Df Sum Sq Mean Sq F value Pr(>F)
## block 6 1114.3 185.71 8.814 0.00358 **
## Treatment 6 1317.4 219.57 10.420 0.00205 **
## Residuals 8 168.6 21.07
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
a) H0: All population means are equal @ no treatments effect
H1: At least one of the population means is different @ there is treatment effects
p-value=0.0021
b) Since p-value=0.0021<alpha=0.05, reject H0.
c) At alpha=0.05, At least one of the population means is different @ there is treatment effects
d) Comparison of treatments
Tukey’s test
TukeyHSD(results)
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Strength ~ block + Treatment, data = datalab3)
##
## $block
## diff lwr upr p adj
## 2-1 7.000000e+00 -7.309004 21.309004 0.5452172
## 3-1 -5.000000e+00 -19.309004 9.309004 0.8200632
## 4-1 1.800000e+01 3.690996 32.309004 0.0145450
## 5-1 -4.263256e-14 -14.309004 14.309004 1.0000000
## 6-1 -3.000000e+00 -17.309004 11.309004 0.9782633
## 7-1 -1.000000e+00 -15.309004 13.309004 0.9999467
## 3-2 -1.200000e+01 -26.309004 2.309004 0.1115313
## 4-2 1.100000e+01 -3.309004 25.309004 0.1575900
## 5-2 -7.000000e+00 -21.309004 7.309004 0.5452172
## 6-2 -1.000000e+01 -24.309004 4.309004 0.2210906
## 7-2 -8.000000e+00 -22.309004 6.309004 0.4149066
## 4-3 2.300000e+01 8.690996 37.309004 0.0032020
## 5-3 5.000000e+00 -9.309004 19.309004 0.8200632
## 6-3 2.000000e+00 -12.309004 16.309004 0.9972844
## 7-3 4.000000e+00 -10.309004 18.309004 0.9217599
## 5-4 -1.800000e+01 -32.309004 -3.690996 0.0145450
## 6-4 -2.100000e+01 -35.309004 -6.690996 0.0057301
## 7-4 -1.900000e+01 -33.309004 -4.690996 0.0105800
## 6-5 -3.000000e+00 -17.309004 11.309004 0.9782633
## 7-5 -1.000000e+00 -15.309004 13.309004 0.9999467
## 7-6 2.000000e+00 -12.309004 16.309004 0.9972844
##
## $Treatment
## diff lwr upr p adj
## 4-2 3.000000 -11.3090041 17.3090041 0.9782633
## 6-2 11.666667 -2.6423374 25.9756707 0.1252127
## 8-2 18.000000 3.6909959 32.3090041 0.0145450
## 10-2 20.333333 6.0243293 34.6423374 0.0070052
## 12-2 5.666667 -8.6423374 19.9756707 0.7332911
## 14-2 9.000000 -5.3090041 23.3090041 0.3060975
## 6-4 8.666667 -5.6423374 22.9756707 0.3397085
## 8-4 15.000000 0.6909959 29.3090041 0.0394390
## 10-4 17.333333 3.0243293 31.6423374 0.0180585
## 12-4 2.666667 -11.6423374 16.9756707 0.9877953
## 14-4 6.000000 -8.3090041 20.3090041 0.6867666
## 8-6 6.333333 -7.9756707 20.6423374 0.6393680
## 10-6 8.666667 -5.6423374 22.9756707 0.3397085
## 12-6 -6.000000 -20.3090041 8.3090041 0.6867666
## 14-6 -2.666667 -16.9756707 11.6423374 0.9877953
## 10-8 2.333333 -11.9756707 16.6423374 0.9938365
## 12-8 -12.333333 -26.6423374 1.9756707 0.0993180
## 14-8 -9.000000 -23.3090041 5.3090041 0.3060975
## 12-10 -14.666667 -28.9756707 -0.3576626 0.0442095
## 14-10 -11.333333 -25.6423374 2.9756707 0.1405131
## 14-12 3.333333 -10.9756707 17.6423374 0.9644345
plot(TukeyHSD(results))
Based on the output, the significants are 8-2, 10-2, 8-4, 10-4, 12-10 because it has p-value < α=0.05.
The most significant comparison is 10-2 because it has the smallest p-value= 0.0070 and the largest difference.