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quantitative-methods/statistical-inference.org

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#+TITLE: Introduction to Statistical Inference
#+PROPERTY: header-args:R :session acj :eval never-export
#+STARTUP: hideall inlineimages hideblocks
#+HTML_HEAD: <style>#content{max-width:1200px;} </style>
* Title slide :slide:
#+BEGIN_SRC emacs-lisp-slide
(org-show-animate '("Quantitative Methods, Part-II" "Introduction to Statistical Inference" "Vikas Rawal" "Prachi Bansal" "" "" ""))
#+END_SRC
* Sampling Distributions
** Sampling Distributions :slide:
# #+RESULTS: sampling2
#+NAME: sampling2
#+BEGIN_SRC R :results output graphics :exports results :file bsample2.png :width 2500 :height 1500 :res 300
library(data.table)
readRDS("plfsdata/plfsacjdata.rds")->worker
worker$standardwage->worker$wage
#read.table("~/ssercloud/acj2018/worker.csv",sep=",",header=T)->worker
c(1:nrow(worker))->worker$SamplingFrameOrder
worker[sex!=3,]->worker
library(ggplot2)
ggplot(worker,aes(wage))+geom_density(colour="black",size=1)+scale_y_continuous(limits=c(0,0.05))+scale_x_continuous(limits=c(0,600),breaks=c(0,mean(worker$wage),1000))->p
# p+facet_wrap(~sex)->p
p+annotate("text",x=380,y=0.045,
label=paste("Population mean = ",round(mean(worker$wage)),sep=""))->p
p+annotate("text",x=400,y=0.042,
label="Distribution of sample means:")->p
p+theme_bw()->p
p
sample(1:nrow(worker),5, replace=FALSE)->a1
worker[a1,]->s1
mean(s1$wage)->t1
for (i in c(1:9999)) {
sample(1:nrow(worker),5, replace=FALSE)->a1
worker[a1,]->s1
c(t1,mean(s1$wage))->t1
}
data.frame(sno=c(1:10000),meancol=t1)->t1
p+geom_density(data=t1,aes(meancol),colour="blue",size=1)-> p
paste("Sample size 5: mean = ",
round(mean(t1$meancol)),
"; stdev = ",
round(sd(t1$meancol)),sep="")->lab
p+annotate("text",x=450,y=0.030,label=lab,colour="blue")->p
p
sample(1:nrow(worker),20, replace=FALSE)->a1
worker[a1,]->s1
mean(s1$wage)->t0
for (i in c(1:9999)) {
sample(1:nrow(worker),20, replace=FALSE)->a1
worker[a1,]->s1
c(t0,mean(s1$wage))->t0
}
data.frame(sno=c(1:10000),meancol=t0)->t0
p+geom_density(data=t0,aes(meancol),colour="darkolivegreen",size=1)-> p
paste("Sample size 20: mean = ",
round(mean(t0$meancol)),
"; stdev = ",
round(sd(t0$meancol)),sep="")->lab
p+annotate("text",x=450,y=0.033,label=lab,colour="darkolivegreen")->p
p
sample(1:nrow(worker),50, replace=FALSE)->a1
worker[a1,]->s1
mean(s1$wage)->t
for (i in c(1:9999)) {
sample(1:nrow(worker),50, replace=FALSE)->a1
worker[a1,]->s1
c(t,mean(s1$wage))->t
}
data.frame(sno=c(1:10000),meancol=t)->t
p+geom_density(data=t,aes(meancol),colour="red",size=1)-> p
paste("Sample size 50: mean = ",
round(mean(t$meancol)),
"; stdev = ",
round(sd(t$meancol)),sep="")->lab
p+annotate("text",x=450,y=0.036,label=lab,colour="red")->p
p
sample(1:nrow(worker),200, replace=FALSE)->a1
worker[a1,]->s1
mean(s1$wage)->t4
for (i in c(1:9999)) {
sample(1:nrow(worker),200, replace=FALSE)->a1
worker[a1,]->s1
c(t4,mean(s1$wage))->t4
}
data.frame(sno=c(1:10000),meancol=t4)->t4
p+geom_density(data=t4,aes(meancol),colour="pink",size=1)-> p
paste("Sample size 200: mean = ",
round(mean(t4$meancol)),
"; stdev = ",
round(sd(t4$meancol)),sep="")->lab
p+annotate("text",x=450,y=0.039,label=lab,colour="pink")->p
p
#+end_src
** Sampling Distributions :slide:
+ $Standard.error = \frac{\sigma}{\sqrt{n}}$
| Variable | Value |
|---------------------------------------------+-------|
| Standard deviation of population ($\sigma$) | 130 |
| Standard errors of samples of size | |
| 5 | 58 |
| 20 | 29 |
| 50 | 18 |
| 200 | 9 |
* Introduction to Hypothesis Testing
** Transforming the Distribution to Standard Normal :slide:
#+RESULTS: sampling3
[[file:bsample3.png]]
#+NAME: sampling3
#+BEGIN_SRC R :results output graphics :exports results :file bsample3.png :width 2500 :height 2000 :res 300
library(data.table)
readRDS("plfsdata/plfsacjdata.rds")->worker
worker$standardwage->worker$wage
c(1:nrow(worker))->worker$SamplingFrameOrder
worker[sex!=3,]->worker
library(ggplot2)
worker->t9
(t9$wage-mean(t9$wage))/sd(t9$wage)->t9$wage
ggplot(t9,aes(wage))+geom_density(colour="black",size=1)->p
p+scale_y_continuous(limits=c(0,0.75))->p
p+scale_x_continuous(limits=c(-15,15)
,breaks=c(-5,0,mean(worker$wage),10,15))->p
p+theme_bw()->p
p
sample(1:nrow(worker),5, replace=FALSE)->a1
worker[a1,]->s1
mean(s1$wage)->t1
for (i in c(1:9999)) {
sample(1:nrow(worker),5, replace=FALSE)->a1
worker[a1,]->s1
c(t1,mean(s1$wage))->t1
}
data.frame(sno=c(1:10000),meancol=(t1-mean(worker$wage))/sd(t1))->t1
p+geom_density(data=t1,aes(meancol),colour="blue",size=1)-> p
p
sample(1:nrow(worker),20, replace=FALSE)->a1
worker[a1,]->s1
mean(s1$wage)->t0
for (i in c(1:9999)) {
sample(1:nrow(worker),20, replace=FALSE)->a1
worker[a1,]->s1
c(t0,mean(s1$wage))->t0
}
data.frame(sno=c(1:10000),meancol=(t0-mean(worker$wage))/sd(t0))->t0
p+geom_density(data=t0,aes(meancol),colour="darkolivegreen",size=1)-> p
p
sample(1:nrow(worker),50, replace=FALSE)->a1
worker[a1,]->s1
mean(s1$wage)->t
for (i in c(1:9999)) {
sample(1:nrow(worker),50, replace=FALSE)->a1
worker[a1,]->s1
c(t,mean(s1$wage))->t
}
data.frame(sno=c(1:10000),meancol=(t-mean(worker$wage))/sd(t))->t
p+geom_density(data=t,aes(meancol),colour="red",size=1)-> p
p
sample(1:nrow(worker),200, replace=FALSE)->a1
worker[a1,]->s1
mean(s1$wage)->t4
for (i in c(1:9999)) {
sample(1:nrow(worker),200, replace=FALSE)->a1
worker[a1,]->s1
c(t4,mean(s1$wage))->t4
}
data.frame(sno=c(1:10000),meancol=(t4-mean(worker$wage))/sd(t4))->t4
p+geom_density(data=t4,aes(meancol),colour="pink",size=1)-> p
p
#+end_src
** Distribution of sample mean with unknown population variance :slide:
#+RESULTS: sampling5
[[file:bsample5.png]]
#+NAME: sampling5
#+BEGIN_SRC R :results output graphics :exports results :file bsample5.png :width 3500 :height 2000 :res 300
library(data.table)
library(ggplot2)
options(scipen=9999)
readRDS("plfsdata/plfsacjdata.rds")->worker
worker$standardwage->worker$wage
c(1:nrow(worker))->worker$SamplingFrameOrder
worker[sex!=3,]->worker
worker->t9
(t9$wage-mean(t9$wage))/sd(t9$wage)->t9$wage
ggplot(t9,aes(wage))+geom_density(colour="black",size=1)->p
p+scale_y_continuous(limits=c(0,0.75))->p
p+scale_x_continuous(limits=c(-15,15)
,breaks=c(-15,0,round(mean(worker$wage)),15))->p
p+theme_bw()->p
p
data.frame(sno=c(),meancol=c(),sterr=c())->t4
samplesize=10
for (i in c(1:20000)) {
sample(1:nrow(worker),samplesize, replace=FALSE)->a1
worker[a1,]->s1
rbind(t4,data.frame(
sno=i,
meancol=mean(s1$wage),
sterr=sd(s1$wage)/sqrt(samplesize)
)
)->t4
}
(t4$meancol)/t4$sterr->t4$teststat
(t4$meancol)/sd(t4$meancol)->t4$teststat2
data.frame(modelt=rt(200000,samplesize-1,ncp=mean(t4$teststat)),modelnorm=rnorm(200000,mean=mean(t4$teststat2)))->m
sd(t4$teststat)
sd(m$modelt)
sd(m$modelnorm)
sd(t4$teststat2)
mean(t4$teststat)
mean(m$modelt)
mean(m$modelnorm)
mean(t4$teststat2)
ggplot()->p
p+geom_density(data=t4,aes(teststat2),colour="red",size=1)-> p
p+geom_density(data=m,aes(modelnorm),colour="black",size=1)->p
p+geom_density(data=t4,aes(teststat),colour="blue",size=1)-> p
p+geom_density(data=m,aes(modelt),colour="darkolivegreen",size=1)->p
p+annotate("text",x=-30,y=0.42,
label=paste("Normal distribution, with standard deviation",round(sd(m$modelnorm),2)),
colour="black",hjust=0)->p
p+annotate("text",x=-30,y=0.40,
label=paste("Statistic with known population variance, standard error =",
round(sd(t4$teststat2),2)),
colour="red",hjust=0)->p
p+annotate("text",x=-30,y=0.38,
label=paste("t distribution, with standard deviation =",round(sd(m$modelt),2)),
colour="darkolivegreen",hjust=0)->p
p+annotate("text",x=-30,y=0.36,
label=paste("Statistic with unknown population variance, standard error =",
round(sd(t4$teststat),2)),
colour="blue",hjust=0)->p
p+scale_x_continuous(limits=c(-30,30))+theme_bw()->p
p
#+end_src
** T Test for means :slide:
*** Testing if the mean is different from a specified value (say zero) :slide:
$H_{0}: \mu = 0$
$H_{a}: \mu \neq 0$
#+name: ttest1
#+begin_src R :results output list org
readRDS("plfsdata/plfsacjdata.rds")->worker
worker$standardwage->worker$wage
worker->t9
t.test(t9$wage)
#+end_src
#+RESULTS: ttest1
#+begin_src org
- One Sample t-test
- data: t9$wage
- t = 432.99, df = 37634, p-value < 0.00000000000000022
- alternative hypothesis: true mean is not equal to 0
- 95 percent confidence interval:
- 289.7136 292.3484
- sample estimates:
- mean of x
- 291.031
#+end_src
*** Testing equality of means :slide:
$H_{0}: \mu_{women} = \mu_{men}$
$H_{a}: \mu_{women} \neq \mu_{men}$
#+name: ttest2
#+begin_src R :results output list org
subset(worker,sex!=3)->t9
factor(t9$sex)->t9$sex
t.test(wage~sex,data=t9)
#+end_src
#+RESULTS: ttest2
#+begin_src org
- Error in subset(worker, sex != 3) : object 'worker' not found
- Error in factor(t9$sex) : object 't9' not found
- Error in eval(m$data, parent.frame()) : object 't9' not found
#+end_src
** Z Test for equality of proportions :slide:
$H_{0}: p_{women} = p_{men}$
$H_{a}: p_{women} \neq p_{men}$
#+name: proptest1
#+begin_src R :results output list org
subset(worker,sex!=3)->t9
as.numeric(t9$gen_edu_level)->t9$gen_edu_level
factor(t9$sex)->t9$sex
t9[gen_edu_level>=8,.(schooled=length(fsu)),.(sex)]->a
t9[,.(all=length(fsu)),.(sex)]->b
prop.test(a$schooled,b$all)
#+end_src
#+CAPTION: Results of test for equality of proportions of men and women who have passed secondary school
#+RESULTS: proptest1
#+begin_src org
- 2-sample test for equality of proportions with continuity correction
- data: a$schooled out of b$all
- X-squared = 847.73, df = 1, p-value < 0.00000000000000022
- alternative hypothesis: two.sided
- 95 percent confidence interval:
- -0.1694726 -0.1525728
- sample estimates:
- prop 1 prop 2
- 0.09245986 0.25348253
#+end_src
** F Test for equality of variances :slide:
$H_{0}: \sigma_{women}^{2} = \sigma_{men}^{2}$
$H_{a}: \sigma_{women}^{2} \neq \sigma_{men}^{2}$
#+name: ftest1
#+begin_src R :results output list org
subset(worker,sex!=3)->t9
factor(t9$sex)->t9$sex
var.test(wage~sex,data=t9)
#+end_src
#+RESULTS: ftest1
#+begin_src org
- F test to compare two variances
- data: wage by sex
- F = 1.8352, num df = 30652, denom df = 6975, p-value <
- 0.00000000000000022
- alternative hypothesis: true ratio of variances is not equal to 1
- 95 percent confidence interval:
- 1.768532 1.903506
- sample estimates:
- ratio of variances
- 1.835174
#+end_src
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