CRD ASSIGNMENT

SUBMITTED BY: MAHRAB KABIR

ROLL: FH-010

a

NULL HYPOTHESIS:- ALL MEANS AR EQUAL

ALTERNET HYPOTHESIS:- ATLEAST ONE MEAN IS DIFFERENT

treat<- rep(c("subcompact", "compact", "midsize", "full size"), each= 10)
obs <- c(3, 5, 3, 7, 6, 5, 3, 2, 1, 6, 1, 3, 4, 7,5, 6, 3, 2, 1, 7, 4, 1, 3, 5, 7, 1, 2, 4, 2, 7, 3, 5, 7, 5, 10, 3, 4, 7, 2, 7)
data.frame(treat,obs)

        treat obs
1  subcompact   3
2  subcompact   5
3  subcompact   3
4  subcompact   7
5  subcompact   6
6  subcompact   5
7  subcompact   3
8  subcompact   2
9  subcompact   1
10 subcompact   6
11    compact   1
12    compact   3
13    compact   4
14    compact   7
15    compact   5
16    compact   6
17    compact   3
18    compact   2
19    compact   1
20    compact   7
21    midsize   4
22    midsize   1
23    midsize   3
24    midsize   5
25    midsize   7
26    midsize   1
27    midsize   2
28    midsize   4
29    midsize   2
30    midsize   7
31  full size   3
32  full size   5
33  full size   7
34  full size   5
35  full size  10
36  full size   3
37  full size   4
38  full size   7
39  full size   2
40  full size   7

n <- 10 
a <- 4
N <- a*n 

CF <- sum(obs)^2/N
CF

[1] 714.025

SST <- sum(obs^2) - CF 
SST

[1] 196.975

treat_total <- tapply(obs,treat, sum)
treat_total

   compact  full size    midsize subcompact 
        39         53         36         41 

SS_treat <- sum(treat_total^2/n) - CF
SS_treat

[1] 16.675

SSE <- SST - SS_treat
SSE

[1] 180.3

df1 <- a-1
dft <- N-1 
df2 <- dft - df1 

MSTreat <- SS_treat / df1
MSTreat

[1] 5.558333

MSE <- SSE/df2
MSE 

[1] 5.008333

fo <- MSTreat / MSE
fo 

[1] 1.109817

ANOVA TABLE

  source_of_variation      SS dof       MS        f
1           Treatment  16.675   3 5.558333 1.109817
2               Error 180.300  36 5.008333       NA
3               Total 196.975  39       NA       NA

f_tab <- qf(0.95, df1, df2)
f_tab

[1] 2.866266

if(fo <= f_tab ){ print("WE MAY ACCEPT THE NULL")
}else{print("WE MAY REJECT THE NULL")}

[1] "WE MAY ACCEPT THE NULL"

So there is no evidence to support a claim that the type of car rented affects the length of the rental contract.

b

combn(c("compact", "full size", "midsize","subcompact"),2)

     [,1]        [,2]      [,3]         [,4]        [,5]         [,6]        
[1,] "compact"   "compact" "compact"    "full size" "full size"  "midsize"   
[2,] "full size" "midsize" "subcompact" "midsize"   "subcompact" "subcompact"

treat_mean <- tapply(obs,treat, mean)
treat_mean

   compact  full size    midsize subcompact 
       3.9        5.3        3.6        4.1 

mean <- as.numeric(treat_mean)
mean_diff <- NULL
for(i in 1:(length(mean) - 1)){
  for(j in (i + 1):length(mean)){
    mean_diff <- c(mean_diff, abs(mean[i]-mean[j]))
  }
}
mean_diff

[1] 1.4 0.3 0.2 1.7 1.2 0.5

LSD method for pair-wise comparison

LSD <- qt(0.05, df2, lower.tail =  FALSE) * sqrt(2*MSE/n)
LSD

[1] 1.689704

for (i in 1:6) {
  if (mean_diff[i] > LSD) {
    print(sprintf("Significant difference for pair %d", i))
  } else {
    print(sprintf("NO significant difference for pair %d", i))
  }
}

[1] "NO significant difference for pair 1"
[1] "NO significant difference for pair 2"
[1] "NO significant difference for pair 3"
[1] "Significant difference for pair 4"
[1] "NO significant difference for pair 5"
[1] "NO significant difference for pair 6"

So according to the LSD test there is only significance difference for pair 4 (full size, midsize).

c

One-at-a-time confidence interval

t_tab <- qt(0.05/2, df2, lower.tail =  FALSE)
t_tab

[1] 2.028094

SE <- sqrt(MSE / n)
treat_mean <- tapply(obs, treat, mean)

Upper bound of One-at-a-time confidence interval

upper_bound <- treat_mean + t_tab*SE
upper_bound

   compact  full size    midsize subcompact 
  5.335274   6.735274   5.035274   5.535274 

Lower bound of One-at-a-time confidence interval

lower_bound <- treat_mean - t_tab*SE
lower_bound

   compact  full size    midsize subcompact 
  2.464726   3.864726   2.164726   2.664726 

Simultaneous Confidence Intervals

t_tab2 <- qt(0.05/2*a, df2, lower.tail =  FALSE)
t_tab2

[1] 1.305514

Upper bound of Simultaneous confidence interval

upper_bound <- treat_mean + t_tab2*SE
upper_bound

   compact  full size    midsize subcompact 
  4.823907   6.223907   4.523907   5.023907 

Lower bound of Simultaneous confidence interval

lower_bound <- treat_mean - t_tab2*SE
lower_bound

   compact  full size    midsize subcompact 
  2.976093   4.376093   2.676093   3.176093