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School
University of Waterloo *
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Course
430
Subject
Industrial Engineering
Date
Dec 6, 2023
Type
Pages
9
Uploaded by HighnessJaguar3858
A1
Adrian Lau
2023-09-23
1.10
A single large experiment is much more likely to waste resources collecting irrelevant data that a pilot study
would have weeded out. A pilot study could also lead to a better match between the sample and the target
population.
1
2.27
a)
a
<-
c(
2.7
,
4.6
,
2.6
,
3.0
,
3.2
,
3.8
)
b
<-
c(
4.6
,
3.4
,
2.9
,
3.5
,
4.1
,
5.1
)
n
<-
6
sa
<-
var(a)
sb
<-
var(b)
ma
<-
mean(a)
mb
<-
mean(b)
t
<-
(ma - mb) / sqrt(sa/n + sb/n)
dof
<-
(sa/n + sb/n)ˆ
2
/((sa/n)ˆ
2
/(n
-1
) + (sb/n)ˆ
2
/(n
-1
))
2
* pt(abs(t),dof,
lower.tail =
FALSE)
## [1] 0.2070179
Since we do not know if the variances are equal, a Welch test was performed, granting
t
0
= 1
.
3487
and degrees
of freedom =
9
.
94
which was rounded to 10.
b) the resulting p-value was calculated to be
0
.
207
, providing no evidence against the means being unequal,
meaning there was no evidence for
C
2
F
6
flow rate affecting etch uniformity
c)
f
<-
sa/sb
2
* pf(f,
5
,
5
)
## [1] 0.8689017
The p-value from this f test was 0.8689, providing no evidence against the variances being equal, meaning
there is no evidence that
C
2
F
6
affects wafer-to-wafer variability in etch uniformity.
d)
boxplot(a,b,
names =
c(
125
,
200
),
xlab =
"C2F6 Flow rate"
,
ylab =
"Etch Uniformity"
,
main =
"Etch Uniformi
2
125
200
2.5
3.0
3.5
4.0
4.5
5.0
Etch Uniformity by Flow Rate
C2F6 Flow rate
Etch Uniformity
3
2.29
a)
a
<-
c(
11.176
,
7.089
,
8.097
,
11.739
,
11.291
,
10.759
,
6.467
,
8.315
)
b
<-
c(
5.263
,
6.748
,
7.461
,
7.015
,
8.133
,
7.418
,
3.772
,
8.963
)
n
<-
8
sa
<-
var(a)
sb
<-
var(b)
ma
<-
mean(a)
mb
<-
mean(b)
t
<-
(ma - mb) / sqrt(sa/n + sb/n)
dof
<-
(sa/n + sb/n)ˆ
2
/((sa/n)ˆ
2
/(n
-1
) + (sb/n)ˆ
2
/(n
-1
))
pt(t,
13
,
lower.tail =
FALSE)
## [1] 0.009538865
Since the variances are not assume to be equal, a Welch test was performed.
t
0
was calculated to be
2
.
67
and
degrees of freedom was calculate to be
13
.
22
which was rounded to
13
b) The resulting p value was calculated to be
0
.
0095
providing very strong evidence against the hypothesis
that the mean of the 100c group is greater than or equal to the mean of the 95c group. Thus there is very
strong evidence that baking at a higher temperature produces a lower mean photoresist thickness.
c)
(ma - mb) - sqrt(sa/n + sb/n)*qt(
0.95
,
13
)
## [1] 0.8517505
The resulting confidence interval is
(0
.
8517505
, Infinity
)
. One of the bounds is infinite since a one sided
test was performed. The implication of the interval is that 95% of generated intervals will contain the true
difference in means, given that they are generated from random samples of the same two distributions.
e)
shapiro.test(a)
##
##
Shapiro-Wilk normality test
##
## data:
a
## W = 0.87501, p-value = 0.1686
shapiro.test(b)
##
##
Shapiro-Wilk normality test
##
## data:
b
## W = 0.9348, p-value = 0.5607
Both shapiro tests p-values showed no evidence against the samples being normally distributed.
f)
rejection_region
<-
qt(
0.05
,
13
)
shift
<-
2.5
/sqrt(sa/n + sb/n)
pt(rejection_region + shift,
13
)
## [1] 0.8033499
4
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