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distribution of sample mean with unknown population variance

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Vikas Rawal 6 years ago
parent bfb7d34a26
commit 0d07bc1d38
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acjlecturesday2.org
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@ -44,7 +44,9 @@ Source: [[https://www.nytimes.com/2018/11/02/opinion/the-perversion-of-fiscal-po
#+END_SRC
** Sampling Distributions :slide:
** Sampling Distributions
*** Sampling Distributions :slide:
#+RESULTS: sampling2
[[file:bsample2.png]]
@ -142,4 +144,239 @@ Source: [[https://www.nytimes.com/2018/11/02/opinion/the-perversion-of-fiscal-po
p
#+end_src
*** Sampling Distributions :slide:
+ $Standard.error = \frac{\sigma}{\sqrt{mean}}$
| 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$wage))/sqrt(var(t9$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$wage))/sqrt(var(t1)))->t1
p+geom_density(data=t1,aes(meancol),colour="blue",size=1)-> p
p
sample(1:nrow(worker),20, replace=FALSE)->a1
worker[a1,]->s1
mean(s1$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))/sqrt(var(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$wage))/sqrt(var(t)))->t
p+geom_density(data=t,aes(meancol),colour="red",size=1)-> p
p
sample(1:nrow(worker),200, replace=FALSE)->a1
worker[a1,]->s1
mean(s1$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))/sqrt(var(t4)))->t4
p+geom_density(data=t4,aes(meancol),colour="pink",size=1)-> p
p
#+end_src
*** But in real situations we do not know the 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$wage))/sqrt(var(t9$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$wage),
sterr=sqrt(var(s1$wage))/sqrt(samplesize)
)
)->t4
}
(t4$meancol)/t4$sterr->t4$teststat
(t4$meancol)/sqrt(var(t4$meancol))->t4$teststat2
data.frame(modelt=rt(200000,samplesize-1,ncp=mean(t4$teststat)),modelnorm=rnorm(200000,mean=mean(t4$teststat2)))->m
var(t4$teststat)
var(m$modelt)
var(m$modelnorm)
var(t4$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(sqrt(var(m$modelnorm)),2)),
colour="black",hjust=0)->p
p+annotate("text",x=-30,y=0.40,
label=paste("Statistic with known population variance, standard error =",
round(sqrt(var(t4$teststat2)),2)),
colour="red",hjust=0)->p
p+annotate("text",x=-30,y=0.38,
label=paste("t distribution, with standard deviation =",round(sqrt(var(m$modelt)),2)),
colour="darkolivegreen",hjust=0)->p
p+annotate("text",x=-30,y=0.36,
label=paste("Statistic with unknown population variance, standard error =",
round(sqrt(var(t4$teststat)),2)),
colour="blue",hjust=0)->p
p+scale_x_continuous(limits=c(-30,30))+theme_bw()->p
p
#+end_src
*** Introduction to the t distribution :ignore:
#+RESULTS: sampling4
[[file:bsample4.png]]
#+NAME: sampling4
#+BEGIN_SRC R :results output graphics :exports results :file bsample4.png :width 2500 :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$wage))/sqrt(var(t9$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=50
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$wage),
sterr=sqrt(var(s1$wage))/sqrt(samplesize)))->t4
}
(t4$meancol-mean(t4$meancol))/t4$sterr->t4$teststat
(t4$meancol-mean(t4$meancol))/sqrt(var(t4$meancol))->t4$teststat2
data.frame(modelt=rt(20000,29))->m
var(t4$teststat)
var(m$modelt)
var(t4$teststat2)
ggplot()->p
p+geom_density(data=t4,aes(teststat),colour="blue",size=1)-> p
p+geom_density(data=m,aes(modelt),colour="darkolivegreen",size=1)->p
p+geom_density(data=t4,aes(teststat2),colour="red",size=1)-> p
p+annotate("text",x=3,y=0.4,
label=paste("Var of statistic with unknown variance:",
round(var(t4$teststat),2)),
colour="blue")->p
p+annotate("text",x=3,y=0.39,
label=paste("Var of statistic with known variance:",
round(var(t4$teststat2),2)),
colour="red")->p
p+annotate("text",x=3,y=0.38,
label=paste("Var of t-distribution:",round(var(m$modelt),2)),
colour="darkolivegreen")->p
p
#+end_src

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