Ex. for 15 Nov. Lecture

adhesion.dat

19.8
15.4
11.4
19.5
10.1
18.5
14.1
8.8
14.9
7.9
17.6
13.6
7.5
12.7
16.7
11.9
15.4
11.9
15.8
11.4
15.4
11.4

anova.R

# ANOVA

# Clean up the workspace
rm(list=ls())

# Import the data
# We have life length for rats, subject to different class of Poison:
#  I, II, III
# and Treatment
#  A, B, C, D
# The final argument of our table is tha length of the life. We can 
# directly import data from the internet:
df <- read.table("https://goo.gl/W9CXQV", header=T)

# Understand the problem with boxplots and interaction plot
boxplot(Life ~ Poison, d=df)
boxplot(Life ~ Treat, d=df)

# In this case we are not sure about residual analysis
# thus it is better to perform ana analysis of residuals
interaction.plot(df$Poison, df$Treat, df$Life)

# Let's make a first model for this process.
# We consider that the the Life process as a dependecies
# from the Treat, the Poison and the combination of Treat and Poison
# Converted in formula it means
df.formula <- Life ~ Treat + Poison + Treat:Poison
# And we can create a model
df.lm <- lm(df.formula, data=df)
# And we can make an analysis of residuals (qualitative) to understand
# if we are making a good assumption of the model
qqnorm(rstandard(df.lm))
qqline(rstandard(df.lm))
# residual pplot
plot(rstandard(df.lm))

# Let's try to simplify this model
df.formula.2 <- Life ~ Treat + Poison
# And we can create a model
df.lm.2 <- lm(df.formula.2, data=df)
# And we can make an analysis of residuals (qualitative) to understand
# if we are making a good assumption of the model
qqnorm(rstandard(df.lm.2))
qqline(rstandard(df.lm.2))
# residual pplot
plot(rstandard(df.lm.2))

# Let's try to improve the model
boxplot(1/Life ~ Poison, d=df)
boxplot(1/Life ~ Treat, d=df)

df.formula.3 <- 1/Life ~ Treat + Poison
# And we can create a model
df.lm.3 <- lm(df.formula.3, data=df)
# And we can make an analysis of residuals (qualitative) to understand
# if we are making a good assumption of the model
qqnorm(rstandard(df.lm.3))
qqline(rstandard(df.lm.3))
# residual pplot
plot(rstandard(df.lm.3))

# Let's try with anova analisys
anova(df.lm.3)
anova(df.lm)

cars.dat

60
53
46
44
47
45
53
43
50
57
48
44
47
53
44
51
55
47
55
46
61
51
57
51
58
51
41
47
48
43
47
53
47
48
52
49
43
47
55
59
50
46
41
58
43
55
55
60
48
48
45
44
44
47
56
48
57
53
44
42
48
59
45
61
43
50
49
47
51
47
64
48
54
47
47
49
49
63
47
58
57
61
48
46
48
47
46
46
54
47
58
48
45
51
47
46
42
51
44
51
44
46
48
55
47
45
57
50
44
52
51
56
48
50
54
47
47
62
45
47
49
47
48
45
57
47
46
44
63
47
47
47
47
46
48
46
43
52
56
54
49
48
45
44
58
45
49
45
47
51
53
48
52
45
43
44
44
42
43
57
49
44
46
46
51
45
49
45
42
53
45
47
50
46
50
47
46
47
47
45
46
50
48
45
58
44
64
46
40
47
55
53
45
53
44
52
50
51
41
51
49
56
52
46
40
48
49
48
47
47
59
54
48
60
47
47
48
48
50
44
58
43
52
46
59
52
45
49
49
64
52
46
59
49
49
47
49
45
59
50
60
45
45
45
48
51
49
46
47
58
62
48
63
56
43
46
48
47
53
44
48
47
46
42
58
47
40
53
44
47
49
47
46
57
51
45
55
45
57
47
47
45
50
53
48
53
47
50
60
64
54
48
50
57
45
48
60
50
54
47
46
44
61
43
47
46
44
57
47
60
48
45
56
47
53
65
54
47
44
45
49
49
65
56
47
47
46
60
45
42
47
46
49
42
53
51
47
55
47
60
50
48
45
44
49
44
51
46
52
52
58
52
48
50
51
61
58
45
61
46
46
53
46
50
49
44
52
49
48
44
42
44
48
46
52
47
47
56
50
47
56
50
62
48
46
57
62
62
50
53
45
55
45
55
44
59
49
45
50
42
59
50
47
44
48
53
50
56
57
60
48
51
63
52
44
47
47
46
57
50
51
47
46
46
46
47
44
47
63
52
45
50
45
44
59
57
50
44
40
45
48
45
47
45
51
47
47
48
44
51
48
43
56
44
46
48
48
55
45
61
50
48
47
46
45
42
55
54
47
47
56
53
45
47
48
47
65
43
45
48
44
54
47
52
42
45
46
52
48
58
48
50
47
46
44
56
50
51
59
49
46
51
61
44
45
50
44
60
47
56
46
50
44
45
43
50
45
44
44
44
58
51
52
47
45
41
48
44
45
46
52
47
47
48
45
51
48
45
48
62
54
41
49
56
49
46
50
47
45
48
48
59
50
49
45
45
47
58
46
49
46
48
56
47
47
47
49
47
46
46
45
56
60
43
44
49
43
49
47
47
52
47
47
49
43
64
47
47
44
58
53
47
45
61
45
44
47
46
45
47
52
46
61
47
57
59
48
48
63
46
57
57
47
55
45
49
60
46
52
47
45
46
50
48
45
46
48
52
54
45
49
43
59
45
47
41
47
54
46
46
50
46
48
44
47
42
55
58
46
61
47
47
45
48
47
53
51
55
48
52
48
45
52
57
60
45
43
45
45
44
45
55
43
51
52
53
47
51
50
52
58
47
48
46
44
47
54
49
51
50
46
52
49
47
46
47
46
46
56
46
56
48
45
49
49
52
44
48
55
48
61
47
46
45
45
46
44
48
51
47
48
42
50
45
57
50
46
51
51
45
47
50
48
45
45
45
46
46
46
46
47
61
48
51
51
47
51
46
57
47
42
47
50
50
48
60
61
45
44
45
44
48
47
58
52
42
50
60
50
46
47
46
50
48
51
60
56
54
52
51
52
46
44
51
47
47
47
54
49
43
53
51
54
50
45
53
46
45
50
50
44
58
40
53
50
57
46
59
47
49
47
42
63
44
47
43
42
47
49
61
50
54
60
46
49
45
49
51
47
56
52
46
47
50
47
46
52
46
47
46
47
58
48
60
46
46
52
52
44
47
54
46
55
52
54
59
49
47
53
64
45
60
57
47
50
52
55
45
47
43
45
44
45
55
57
40
62
52
47
44
49
51
49
45
54
60
43
50
42
59
42
48
46
53
47
42
54
49
46
57
47
45
49
54
47
45
45
45
44
46
46
47
58
49
55
50
49
48
52
63
47
47
51
48
45
58
44
56
52
60
47
46
44
44
48
47
49
48
61
56
46
63
44
44
45
52
46
46
49
56
46
51
45
47
42
57
56
46
57
51
49
46
42
44
48
50
43
43
47
51
53
46
65
47
62
57
46
53
56
57
48
50
51
62
58
49
60
41
42
46
48
62
53
51
50
49
46
53
46
57
49
49
59
43
65
45
50
46
47
50
49
60
45
52
46
65
44
48
53
59
50
48
44
55
46
47
45
53
47
43
47
48
47
62
53
48
61
45
55
51
45
59
50
48
46
47
47
52
60
51
53
45
49
61
52
47
53
61
43
63
52
46
51
50
53
52
52
45
44
49
42
49
45
44
45
46
62
47
44
54
51
44
47
51
59
49
46
46
48
47
51
55
43
47
62
43
45
45
55
54
48
47
61
65
46
45
45
53
57
53
46
47
54
62
44
56
44
56
47
47
60
56
46
45
44
56
47
53
43
55
50
58
44
53
48
46
45
47
48
44
45
54
61
62
48
48
48
60
56
49
56
45
44
46
46
58
54
49
55
43
51
46
46
45
45
45
49
44
49
50
45
46
43
65
48
60
52
43
51
50
49
48
50
53
54
49
44
59
43
50
50
43
48
44
48
45
45
57
54
44
65
54
54
46
44
43
52
46
44
51
61
44
52
43
57
46
52
49
45
48
51
56
63
58
43
51
63
49
57
50
45
49
46
48
41
47
46
44
46
47
44
47
53
50
41
56
54
45
48
48
48
44
47
43
54
52
59
58
47
43
43
46
47
43
47
57
62
53
47
48
53
55
58
43
47
46
51
45
47
51
54
42
41
49
65
55
53
47
50
53
44
51
43
45
53
54
46
62
45
47
50
50
51
46
57
45
45
46
54
49
47
58
54
49
40
50
48
47
49
47
49
47
57
48
45
47
49
44
53
62
45
60
56
43
49
51
56
47
50
61
45
44
47
53
64
55
47
54
46
48
43
46
49
50
51
47
47
56
46
60
55
47
56
54
52
54
46
47
49
49
56
57
55
53
54
56
52
50
45
49
58
60
59
42
42
48
53
48
46
53
47
46
50
44
50
44
50
52
43
45
46
47
56
51
51
51
48
51
52
47
61
47
44
63
55
46
47
56
46
56
49
53
45
48
60
50
46
43
48
59
55
52
44
51
57
63
64
47
49
41
48
45
49
46
47
50
46
52
47
47
50
45
45
48
46
47
53
47
50
46
46
49
45
49
46
56
47
47
50
44
57
55
47
50
62
52
54
44
41
44
47
52
44
57
47
49
53
44
54
47
45
43
53
53
47
45
45
52
55
59
58
48
57
46
59
53
64
58
58
47
43
53
53
48
62
52
44
45
50
59
49
61
60
44
43
54
48
45
51
53
53
47
46
50
43
48
65
48
50
57
47
48
48
48
48
47
44
60
43
47
54
46
46
55
48
48
46
51
51
47
51
46
49
54
45
64
47
60
44
47
51
48
49
47
56
48
47
60
47
48
46
57
48
46
49
49
51
49
47
55
50
50
62
49
47
50
49
60
48
56
46
48
53
53
44
50
50
55
49
40
50
55
48
65
45
43
58
44
41
52
55
41
45
45
60
61
43
44
54
44
52
49
48
54
51
53
44
48
51
51
48
59
53
60
46
41
62
47
43
60
47
45
46
59
48
44
52
43
57
46
56
46
60
45
49
43
47
46
50
51
52
41
48
51
45
45
46
48
64
54
50
56
55
49
47
47
50
45
54
63
59
46
48
42
50
48
47
48
58
44
49
45
49
44
46
43
52
47
41
48
46
47
46
45
47
47
49
62
53
61
49
47
46
48
46
59
54
54
46
64
59
48
53
48
45
60
45
43
50
45
56
46
55
50
43
45
44
47
47
48
44
51
48
43
54
59
46
48
47
47
54
47
49
44
45
58
48
47
45
54
51
47
47
48
50
42
50
50
48
54
48
56
47
55
53
53
51
51
51
47
40
46
57
56
54
54
47
50
46
51
52
47
45
49
46
43
40
47
48
49
50
42
42
45
48
52
47
48
47
50
47
46
46
53
43
51
50
47
50
58
46
49
43
47
57
47
48
46
49
47
54
65
52
45
59
47
46
45
55
50
50
52
45
62
50
54
46
43
64
49
57
60
46
46
43
45
50
51
49
53
45
58
54
50
49
59
42
48
47
55
47
49
45
44
60
56
51
46
45
58
46
55
54
43
50
53
49
46
44
52
55
46
48
54
46
55
49
46
45
47
47
58
54
46
49
63
50
65
57
57
54
48
50
57
49
48
49
51
59
61
57
55
45
44
49
55
40
55
48
43
55
52
44
50
58
47
45
56
50
45
51
56
50
51
45
61
49
47
48
62
53
48
44
56
44
52
43
46
57
48
50
47
54
47
46
44
55
55
47
53
50
48
52
49
49
62
58
47
46
48
47
47
60
46
45
56
50
54
47
47
41
53
44
43
61
56
43
43
54
55
54
43
49
45
45
58
48
50
48
52
42
50
49
47
61
51
59
44
47
49
44
50
52
45
46
62
58
51
52
48
43
54
46
49
49
43
44
51
47
46
57
57
47
62
49
51
46
51
47
42
46
46
50
43
60
51
48
46
45
63
46
54
46
47
48
46
44
47
41
48
43
50
48
45
52
46
44
63
44
48
45
56
58
41
48
48
47
59
59
44
58
51
46
47
56
45
63
46
58
54
42
63
59
52
46
61
45
47
59
47
51
54
47
55
53
64
46
59
55
50
47
44
53
45
47
48
46
46
47
44
45
55
49
46
60
51
45
61
46
51
41
49
50
49
46
46
47
51
57
45
59
47
49
49
46
53
44
44
41
50
55
47
44
61
62
45
47
60
59
46
47
48
58
63
48
44
46
48
46
44
48
54
53
52
48
50
44
55
59
46
47
51
49
45
46
61
65
47
45
56
51
60
46
57
51
50
56
46
54
47
47
53
45
45
43
43
46
54
43
45
47
46
52
46
59
47
52
52
52
54
45
40
59
49
53
61
43
56
45
44
63
44
57
44
51
47
43
47
46
41
45
57
54
47
47
48
47
60
64
55
53
44
51
43
49
53
45
48
46
45
47
56
48
49
49
59
55
47
46
54
50
54
48
45
45
45
45
51
50
48
44
45
42
52
45
46
47
42
62
56
44
60
47
62
44
45
46
55
46
47
61
44
45
44
60
46
50
48
56
45
49
49
47
47
47
55
55
45
47
46
51
47
48
48
54
45
55
59
48
47
52
46
62
49
46
52
44
45
58
46
51
53
42
47
56
49
46
49
44
44
63
53
44
46
43
49
47
55
41
52
46
49
47
62
52
47
50
46
45
45
46
44
42
62
49
54
42
47
46
43
45
47
50
49
51
51
42
45
45
47
44
44
51
50
51
47
52
57
47
54
51
51
48
59
60
49
60
47
58
50
50
48
49
45
48
56
59
55
51
44
56
47
54
59
43
46
44
49
50
44
47
62
51
57
60
49
55
62
50
45
51
60
63
52
46
47
47
48
53
57
48
45
48
54
43
47
47
55
49
46
48
46
47
45
47
40
51
48
56
48
45
49
50
43
53
45
54
51
51
51
48
56
46
42
48
47
44
47
45
52
47
49
42
46
49
48
46
45
49
58
57
55
51
51
52
48
44
55
47
50
61
43
55
46
46
46
49
61
60
52
45
45
47
56
42
51
53
45
49
50
46
62
50
44
57
60
42
46
42
52
50
63
44
61
58
54
44
59
50
47
47
46
48
44
46
47
50
46
43
55
62
56
45
47
43
51
59
43
50
55
64
51
46
61
46
44
47
47
46
48
48
48
49
52
49
47
47
51
61
56
61
46
46
49
47
47
55
54
50
44
45
44
59
55
48
42
62
48
52
57
62
61
44
46
47
44
47
48
48
45
58
50
61
46
59
46
61
51
46
56
44
46
47
48
62
52
47
43
53
51
57
45
49
44
43
65
50
49
51
43
63
57
57
43
48
57
54
63
53
43
54
54
46
46
44
60
42
51
45
44
55
45
51
55
47
46
58
47
52
42
45
46
48
43
48
51
44
48
44
47
55
50
48
44
50
46
43
51
49
47
58
48
55
48
48
47
59
49
45
48
47
48
49
46
40
63
50
49
47
53
45
54
62
46
48
52
43
64
44
63
45
47
45
58
47
62
48
44
58
43
45
44
55
50
46
59
53
44
45
50
59
53
58
47
48
61
54
52
42
55
47
55
56
61
44
46
44
48
46
51
47
45
58
46
49
43
45
47
47
51
46
53
45
48
47
49
46
50
42
46
50
55
48
50
45
45
57
61
44
47
50
47
45
48
46
51
49
45
51
47
43
52
54
48
49
49
47
44
53
48
42
47
42
42
49
43
50
46
43
52
46
63
58
47
53
42
58
57
55
45
45
43
49
45
58
47
46
44
54
40
47
46
46
46
51
51
56
49
44
57
61
51
56
56
52
61
58
47
45
44
58
46
47
51
46
42
48
46
55
60
55
49
46
56
51
48
49
50
44
48
52
49
56
50
53
42
49
49
48
48
59
44
44
45
44
44
44
51
48
54
47
50
48
55
57
47
48
41
53
45
47
46
47
53
50
47
57
47
48
45
47
53
49
49
49
48
48
48
51
48
45
48
46
45
48
44
44
52
47
56
46
46
47
49
48
45
53
59
58
48
52
56
45
50
47
51
46
52
50
64
59
45
43
48
64
43
45
47
43
51
45
42

chi.square.R

# GOODNESS OF FIT
# This example represent bservations of vehicle per minute
# in a street. Evaluate if this pcess can be modeled with 
# a Poisson distribution.
rm(list=ls())

# Data are available at https://goo.gl/v91n7L
df <- read.table("https://goo.gl/v91n7L")$V1

df.n <- length(df)
df.m <- mean(df)
df.s <- sd(df)
df.l <- 46

# Create the bins
# We put freq=F to get a density distribution instead of a pure
# frequency distribution
(df.h <- hist(df, freq=F))

# Create the expected distribution
df.b <- df.h$breaks # x-values in hist plot
#df.b[1] = -Inf
df.b[length(df.b)] = 0

# Evaluate the expected Poisson distribution
ei <- diff(ppois(df.b, df.l)) * df.n
plot(ei)
points(df.h$counts, col="red")

chi <-  sum((df.h$counts - ei) ** 2 / ei)

# Check against pchisq
pchisq(chi, df=length(df.h$counts) - 1 - 1)

conf.int.R

# CONFIDENCE INTERVAL

rm(list=ls())

# An article describes t results of tensile adhesion tests on 
# n = 22 U-700 Alloy specimens. The failure load is described in file
# at: https://goo.gl/6XWdME
df <- read.table(file="https://goo.gl/6XWdME")$V1


(m <- mean(df))
(s <- sd(df))
(n <- length(df))

# Let's check our model for a distribution 
boxplot(df, ylab="Load at Failure")

qqnorm(df)
qqline(df)

# We can assume that the population is normally distributed
# and we want to understand which is the confidence interval 
# on the mean of our distribution
# The confidence interval is the confidence interval on the mean
# with variance unknown for the t-distribution
conf <- 0.95
t <- qt((1 - conf)/2, n - 1, low=F) 

(c(
  m - t * s / sqrt(n),
  m + t * s / sqrt(n)
))

rats.dat

Poison Treat Life
I	A	0.31
I	B	0.82
I	C	0.43
I	D	0.45
I	A	0.45
I	B	1.1
I	C	0.45
I	D	0.71
I	A	0.46
I	B	0.88
I	C	0.63
I	D	0.66
I	A	0.43
I	B	0.72
I	C	0.76
I	D	0.62
II	A	0.36
II	B	0.92
II	C	0.44
II	D	0.56
II	A	0.29
II	B	0.61
II	C	0.35
II	D	1.02
II	A	0.4
II	B	0.49
II	C	0.31
II	D	0.71
II	A	0.23
II	B	1.24
II	C	0.4
II	D	0.38
III	A	0.22
III	B	0.3
III	C	0.23
III	D	0.3
III	A	0.21
III	B	0.37
III	C	0.25
III	D	0.36
III	A	0.18
III	B	0.38
III	C	0.24
III	D	0.31
III	A	0.23
III	B	0.29
III	C	0.22
III	D	0.33

t.test.R

# T-Test example - 15 Nov 2016
# Clear workspace
rm(list=ls())

# PART 1
# DATA GENERATION

# Define two means µ[a] and µ[b], suche that µ[a] > µ[b]
mu.a <-  0.1
mu.b <- -0.1

# Define two standard deviation σ[a] and σ[b], such that σ[a]!= σ[b]
sigma.a <- 0.1
sigma.b <- 0.15

# Create 4 groups of samples such that:
#  * N(µa, σa) with 12 samples -> s1
#  * N(µa, σb) with 15 samples -> s2
#  * N(µb, σa) with 13 samples -> s3
#  * N(µb, σb) with 10 samples -> s4
n.1 <- 12
s.1 <- rnorm(n.1, mu.a, sigma.a)

n.2 <- 15
s.2 <- rnorm(n.2, mu.a, sigma.b)

n.3 <- 13
s.3 <- rnorm(n.3, mu.b, sigma.a)

n.4 <- 10
s.4 <- rnorm(n.4, mu.b, sigma.b)

# PART 2
# T-TEST ONE SAMPLE, TWO SIDES
# t.test if s4 belongs to a distribution with mean µ[b]
#  H0: µ1 = µ2
#  H1: µ1 != µ2

# In this case there is no need to evaluate the pooled variance,
# but we can directly evaluate the t0
test.1.t0 <- (mean(s.4) - mu.b)/(sd(s.4) / sqrt(n.4))
# And we get a p-value from a two sided t-distribution
( pt(abs(test.1.t0), n.4 - 1, lower.tail=FALSE) * 2 )
# Lets test it versus built in function
( test.1.tt <- t.test(s.4, alternative="two.sided", mu=mu.b, paired=F) )

# PART 3
# INTERLUDE: Functions
# Write that evaluates t-test for sigma equal and sigma different
my.t.test <- function(s1, s2, sigma.equal=TRUE) {
  n1 <- length(s1)
  n2 <- length(s2)
  
  t0 <- 0
  if (sigma.equal) {
    n  <- n1 + n2 - 2
    sp <- ((n1-1)*var(s1) + (n2-1)*var(s2)) / n
    return((mean(s1)-mean(s2))/(sqrt(sp*(1/n1+1/n2))))
  } else {
    return((mean(s1) - mean(s2))/sqrt(var(s1)/n1 + var(s2)/n2))  
  }
}

# PART 4
# T-TEST TWO SAMPLES, TWO SIDES
# t.test between s1 and s2 with
#  H0: µ1 = µ2
#  H1: µ1 != µ2
test.2.t0 <- my.t.test(s.1, s.2)
( pt(abs(test.2.t0), (n.1 + n.2 - 2), lower.tail=FALSE) * 2 )
( test.2.tt <- t.test(s.1, y=s.2, alternative="two.sided", var.equal=T) )

# PART 5
# T-TEST TWO SAMPLES, ONE SIDE
# t.test between s1 and s3 with
#  H0: µ1 = µ2
#  H1: µ1 < µ2
test.3.t0 <- my.t.test(s.1, s.3)
( pt(test.3.t0, n.1 + n.3 - 2, low=T) )
( test.3.tt <- t.test(s.1, y=s.3, alt="less", var.equal=T))


# PART 6
# T-TEST TWO SAMPLES, ONE SIDE
# t.test between s2 and s4 with
#  H0: µ1 = µ2
#  H1: µ1 > µ2
test.4.t0 <- my.t.test(s.2, s.4)
( pt(test.4.t0, n.2 + n.4 - 2, low=F) )
( test.4.tt <- t.test(s.2, y=s.4, paired=F, alt="greater", var.equal=T))

# HOMEWORK
# Imagine that you now know a priori:
#
#  * s.1 deviation differs from s.2 deviation
#  * s.1 deviation is the same of s.3 deviation
#  * s.2 deviation is the same of s.4 deviation
#
# Modify the script to make a "good" comparison, using 
# var.equal=FALSE where it is necessary.
# Write a function to evaluate the degree of freedom
# that is necessary in that case.
# hint: Welch-Satterthwaite equation

# PART 7
# Evaluate the confidence interval for each of the previous 
# two-samples comparison
pool.deviation <- function(s1, s2) {
  n1 <- length(s1)
  n2 <- length(s2)
  n  <- n1 + n2 - 2
  sp <- (n1-1)*var(s1) + (n2-1)*var(s2)
  return(sqrt(sp/n))
}

scaling.coefficient <- function(s1, s2, var.equal=T) {
  n1 <- length(s1)
  n2 <- length(s2)
  
  if(var.equal) {
    sp <- pool.deviation(s1, s2)
    sn <- sqrt((1/n1) + (1/n2))
    return(sp*sn)
  } else {
    return(sqrt((var(s1)/n1) + (var(s2)/n2)))
  }
}

confidence.interval <- function(s1, s2, conf=0.95, var.equal=T) {
  dmu   <- mean(s1) - mean(s2)
  
  n <- 0
  if (var.equal) {  
    n <- length(s1) + length(s2) - 2
  } else {
    # To be completed with a paricular equation...
    # this means... HOMEWORK!! yup!
    n <- 0 # ?
  }
  
  cf    <- scaling.coefficient(s1, s2, var.equal=var.equal)
  tp    <- qt((1 - conf)/2, n, low=F)
  
  i_min <- dmu - tp * cf
  i_max <- dmu + tp * cf

  return(c(i_min, i_max))
} 

( test.2.tt$conf.int )
( confidence.interval(s.1, s.2) )