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329 lines
9.8 KiB
329 lines
9.8 KiB
#+TITLE: Quantitative Methods
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#+PROPERTY: header-args:R :session acj :eval never-export
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#+STARTUP: hideall inlineimages hideblocks
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#+HTML_HEAD: <style>#content{max-width:1200px;} </style>
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* Title slide :slide:
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#+BEGIN_SRC emacs-lisp-slide
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(org-show-animate '("Quantitative Methods, Part-II" "Introduction to Statistical Inference" "Vikas Rawal" "Prachi Bansal" "" "" ""))
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#+END_SRC
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* Sampling Distributions
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** Sampling Distributions :slide:
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# #+RESULTS: sampling2
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[[file:bsample2.png]]
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#+NAME: sampling2
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#+BEGIN_SRC R :results output graphics :exports results :file bsample2.png :width 2500 :height 1500 :res 300
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library(data.table)
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readRDS("plfsdata/plfsacjdata.rds")->worker
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worker$standardwage->worker$wage
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#read.table("~/ssercloud/acj2018/worker.csv",sep=",",header=T)->worker
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c(1:nrow(worker))->worker$SamplingFrameOrder
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worker[sex!=3,]->worker
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library(ggplot2)
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ggplot(worker,aes(wage))+geom_density(colour="black",size=1)+scale_y_continuous(limits=c(0,0.05))+scale_x_continuous(limits=c(0,600),breaks=c(0,mean(worker$wage),1000))->p
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# p+facet_wrap(~sex)->p
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p+annotate("text",x=380,y=0.045,
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label=paste("Population mean = ",round(mean(worker$wage)),sep=""))->p
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p+annotate("text",x=400,y=0.042,
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label="Distribution of sample means:")->p
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p+theme_bw()->p
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p
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sample(1:nrow(worker),5, replace=FALSE)->a1
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worker[a1,]->s1
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mean(s1$wage)->t1
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for (i in c(1:9999)) {
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sample(1:nrow(worker),5, replace=FALSE)->a1
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worker[a1,]->s1
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c(t1,mean(s1$wage))->t1
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}
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data.frame(sno=c(1:10000),meancol=t1)->t1
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p+geom_density(data=t1,aes(meancol),colour="blue",size=1)-> p
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paste("Sample size 5: mean = ",
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round(mean(t1$meancol)),
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"; stdev = ",
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round(sd(t1$meancol)),sep="")->lab
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p+annotate("text",x=450,y=0.030,label=lab,colour="blue")->p
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p
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sample(1:nrow(worker),20, replace=FALSE)->a1
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worker[a1,]->s1
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mean(s1$wage)->t0
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for (i in c(1:9999)) {
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sample(1:nrow(worker),20, replace=FALSE)->a1
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worker[a1,]->s1
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c(t0,mean(s1$wage))->t0
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}
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data.frame(sno=c(1:10000),meancol=t0)->t0
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p+geom_density(data=t0,aes(meancol),colour="darkolivegreen",size=1)-> p
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paste("Sample size 20: mean = ",
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round(mean(t0$meancol)),
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"; stdev = ",
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round(sd(t0$meancol)),sep="")->lab
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p+annotate("text",x=450,y=0.033,label=lab,colour="darkolivegreen")->p
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p
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sample(1:nrow(worker),50, replace=FALSE)->a1
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worker[a1,]->s1
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mean(s1$wage)->t
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for (i in c(1:9999)) {
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sample(1:nrow(worker),50, replace=FALSE)->a1
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worker[a1,]->s1
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c(t,mean(s1$wage))->t
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}
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data.frame(sno=c(1:10000),meancol=t)->t
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p+geom_density(data=t,aes(meancol),colour="red",size=1)-> p
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paste("Sample size 50: mean = ",
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round(mean(t$meancol)),
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"; stdev = ",
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round(sd(t$meancol)),sep="")->lab
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p+annotate("text",x=450,y=0.036,label=lab,colour="red")->p
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p
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sample(1:nrow(worker),200, replace=FALSE)->a1
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worker[a1,]->s1
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mean(s1$wage)->t4
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for (i in c(1:9999)) {
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sample(1:nrow(worker),200, replace=FALSE)->a1
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worker[a1,]->s1
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c(t4,mean(s1$wage))->t4
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}
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data.frame(sno=c(1:10000),meancol=t4)->t4
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p+geom_density(data=t4,aes(meancol),colour="pink",size=1)-> p
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paste("Sample size 200: mean = ",
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round(mean(t4$meancol)),
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"; stdev = ",
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round(sd(t4$meancol)),sep="")->lab
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p+annotate("text",x=450,y=0.039,label=lab,colour="pink")->p
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p
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#+end_src
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** Sampling Distributions :slide:
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+ $Standard.error = \frac{\sigma}{\sqrt{mean}}$
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| Variable | Value |
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|---------------------------------------------+-------|
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| Standard deviation of population ($\sigma$) | 130 |
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| Standard errors of samples of size | |
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| 5 | 58 |
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| 20 | 29 |
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| 50 | 18 |
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| 200 | 9 |
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* Introduction to Hypothesis Testing
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** Transforming the Distribution to Standard Normal :slide:
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#+RESULTS: sampling3
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[[file:bsample3.png]]
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#+NAME: sampling3
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#+BEGIN_SRC R :results output graphics :exports results :file bsample3.png :width 2500 :height 2000 :res 300
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library(data.table)
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readRDS("plfsdata/plfsacjdata.rds")->worker
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worker$standardwage->worker$wage
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c(1:nrow(worker))->worker$SamplingFrameOrder
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worker[sex!=3,]->worker
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library(ggplot2)
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worker->t9
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(t9$wage-mean(t9$wage))/sd(t9$wage)->t9$wage
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ggplot(t9,aes(wage))+geom_density(colour="black",size=1)->p
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p+scale_y_continuous(limits=c(0,0.75))->p
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p+scale_x_continuous(limits=c(-15,15)
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,breaks=c(-5,0,mean(worker$wage),10,15))->p
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p+theme_bw()->p
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p
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sample(1:nrow(worker),5, replace=FALSE)->a1
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worker[a1,]->s1
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mean(s1$wage)->t1
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for (i in c(1:9999)) {
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sample(1:nrow(worker),5, replace=FALSE)->a1
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worker[a1,]->s1
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c(t1,mean(s1$wage))->t1
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}
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data.frame(sno=c(1:10000),meancol=(t1-mean(worker$wage))/sd(t1))->t1
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p+geom_density(data=t1,aes(meancol),colour="blue",size=1)-> p
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p
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sample(1:nrow(worker),20, replace=FALSE)->a1
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worker[a1,]->s1
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mean(s1$wage)->t0
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for (i in c(1:9999)) {
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sample(1:nrow(worker),20, replace=FALSE)->a1
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worker[a1,]->s1
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c(t0,mean(s1$wage))->t0
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}
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data.frame(sno=c(1:10000),meancol=(t0-mean(worker$wage))/sd(t0))->t0
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p+geom_density(data=t0,aes(meancol),colour="darkolivegreen",size=1)-> p
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p
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sample(1:nrow(worker),50, replace=FALSE)->a1
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worker[a1,]->s1
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mean(s1$wage)->t
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for (i in c(1:9999)) {
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sample(1:nrow(worker),50, replace=FALSE)->a1
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worker[a1,]->s1
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c(t,mean(s1$wage))->t
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}
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data.frame(sno=c(1:10000),meancol=(t-mean(worker$wage))/sd(t))->t
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p+geom_density(data=t,aes(meancol),colour="red",size=1)-> p
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p
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sample(1:nrow(worker),200, replace=FALSE)->a1
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worker[a1,]->s1
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mean(s1$wage)->t4
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for (i in c(1:9999)) {
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sample(1:nrow(worker),200, replace=FALSE)->a1
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worker[a1,]->s1
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c(t4,mean(s1$wage))->t4
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}
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data.frame(sno=c(1:10000),meancol=(t4-mean(worker$wage))/sd(t4))->t4
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p+geom_density(data=t4,aes(meancol),colour="pink",size=1)-> p
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p
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#+end_src
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** Distribution of sample mean with unknown population variance :slide:
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#+RESULTS: sampling5
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[[file:bsample5.png]]
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#+NAME: sampling5
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#+BEGIN_SRC R :results output graphics :exports results :file bsample5.png :width 3500 :height 2000 :res 300
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library(data.table)
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library(ggplot2)
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options(scipen=9999)
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readRDS("plfsdata/plfsacjdata.rds")->worker
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worker$standardwage->worker$wage
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c(1:nrow(worker))->worker$SamplingFrameOrder
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worker[sex!=3,]->worker
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worker->t9
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(t9$wage-mean(t9$wage))/sd(t9$wage)->t9$wage
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ggplot(t9,aes(wage))+geom_density(colour="black",size=1)->p
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p+scale_y_continuous(limits=c(0,0.75))->p
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p+scale_x_continuous(limits=c(-15,15)
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,breaks=c(-15,0,round(mean(worker$wage)),15))->p
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p+theme_bw()->p
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p
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data.frame(sno=c(),meancol=c(),sterr=c())->t4
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samplesize=10
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for (i in c(1:20000)) {
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sample(1:nrow(worker),samplesize, replace=FALSE)->a1
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worker[a1,]->s1
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rbind(t4,data.frame(
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sno=i,
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meancol=mean(s1$wage),
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sterr=sd(s1$wage)/sqrt(samplesize)
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)
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)->t4
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}
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(t4$meancol)/t4$sterr->t4$teststat
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(t4$meancol)/sd(t4$meancol)->t4$teststat2
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data.frame(modelt=rt(200000,samplesize-1,ncp=mean(t4$teststat)),modelnorm=rnorm(200000,mean=mean(t4$teststat2)))->m
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sd(t4$teststat)
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sd(m$modelt)
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sd(m$modelnorm)
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sd(t4$teststat2)
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mean(t4$teststat)
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mean(m$modelt)
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mean(m$modelnorm)
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mean(t4$teststat2)
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ggplot()->p
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p+geom_density(data=t4,aes(teststat2),colour="red",size=1)-> p
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p+geom_density(data=m,aes(modelnorm),colour="black",size=1)->p
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p+geom_density(data=t4,aes(teststat),colour="blue",size=1)-> p
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p+geom_density(data=m,aes(modelt),colour="darkolivegreen",size=1)->p
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p+annotate("text",x=-30,y=0.42,
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label=paste("Normal distribution, with standard deviation",round(sd(m$modelnorm),2)),
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colour="black",hjust=0)->p
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p+annotate("text",x=-30,y=0.40,
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label=paste("Statistic with known population variance, standard error =",
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round(sd(t4$teststat2),2)),
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colour="red",hjust=0)->p
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p+annotate("text",x=-30,y=0.38,
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label=paste("t distribution, with standard deviation =",round(sd(m$modelt),2)),
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colour="darkolivegreen",hjust=0)->p
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p+annotate("text",x=-30,y=0.36,
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label=paste("Statistic with unknown population variance, standard error =",
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round(sd(t4$teststat),2)),
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colour="blue",hjust=0)->p
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p+scale_x_continuous(limits=c(-30,30))+theme_bw()->p
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p
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#+end_src
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** t-test
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#+name: ttest1
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#+begin_src R :results output
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readRDS("plfsdata/plfsacjdata.rds")->worker
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worker$standardwage->worker$wage
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t.test(worker$wage)
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#+end_src
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#+RESULTS: ttest1
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#+begin_example
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One Sample t-test
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data: worker$wage
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t = 432.99, df = 37634, p-value < 0.00000000000000022
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alternative hypothesis: true mean is not equal to 0
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95 percent confidence interval:
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289.7136 292.3484
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sample estimates:
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mean of x
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291.031
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#+end_example
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#+name: ttest2
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#+begin_src R :results output
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subset(worker,sex!=3)->t9
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factor(t9$sex)->t9$sex
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t.test(wage~sex,data=t9)
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#+end_src
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#+RESULTS: ttest2
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#+begin_example
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Welch Two Sample t-test
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data: wage by sex
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t = 79.02, df = 13483, p-value < 0.00000000000000022
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alternative hypothesis: true difference in means is not equal to 0
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95 percent confidence interval:
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104.6563 109.9805
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sample estimates:
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mean in group 1 mean in group 2
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310.8974 203.5790
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#+end_example
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