Set dictionary and Import data Set

setwd("~/Desktop")
hincp<- read.csv("ACS_2020_NY_subset.csv",h=T)
head(hincp)
##        serialno  hincp
## 1 2020HU0000013   7300
## 2 2020HU0000022    900
## 3 2020HU0000038  55900
## 4 2020HU0000040  35800
## 5 2020HU0000051 202000
## 6 2020HU0000055  99200
names(hincp)
## [1] "serialno" "hincp"
dim(hincp)
## [1] 60360     2

Question # 1

Histogram

histogram<-hist(hincp$hincp,main="Household Income Distribution",xlab="Household Income $")

Interpretation

We can clearly observe most household income lies left sided of graph so we can say that it is left or positively skewed

QQ plot

library(car)
## Loading required package: carData
qqnorm(hincp$hincp, pch = 1, frame = FALSE)
qqline(hincp$hincp, col = "red", lwd = 2)

Interpretation

Mostly data lies above the line so we can say the household income does not follow the normal distribution it is positive skewed.

Question # 2

population_mean<-mean(hincp$hincp)
population_mean
## [1] 110719.6
population_sd<-sd(hincp$hincp)
population_sd
## [1] 126177.7

Interpretation

The all household average income is 110719.

Question 3

Draw 10,000 random samples of size n = 2 from the population and:

a. Create a histogram of the sampling distribution (hint: first you need to calculate the sample mean in each sample)

#make this example reproducible

sample<-sample(1:10000,2,replace=T)
sample_means<-mean(sample)
head(sample_means)
## [1] 5723.5
#Create a histogram of the sampling distribution
hist(sample, main = " ", xlab = "Sample Means", col = "steelblue")

Interpretation

Graph shows normal distribution

b. Create a Q-Q plot of the sampling distribution to check its normality;

qqnorm(sample, pch = 1, frame = FALSE)
qqline(sample, col = "red", lwd = 2)

Interpretation

Graph shows normal distribution

c. Compute the mean of the sampling distribution and compare it to the “population mean”.

Samples_Mean<-mean(sample_means)
Samples_Mean
## [1] 5723.5
population_mean<-mean(hincp$hincp)
population_mean
## [1] 110719.6

Comparison

We can clearly observe that the sample mean is much less than the population mean.

Question 4

Draw 10,000 random samples of size n = 10 from the population and:

a. Create a histogram of the sampling distribution (hint: first you need to calculate the sample mean in each sample);

#make this example reproducible

set.seed(0)

#define number of samples 

sample<-sample(1:10000,10,replace=T)
sample_means<-mean(sample)
head(sample_means)
## [1] 6425.9
#view first six sample means
hist(sample, main = "", xlab = "Sample Means", col = "steelblue")

Interpretation

Graph shows not normal distribution

b. Create a Q-Q plot of the sampling distribution to check its normality;

qqnorm(sample, pch = 1, frame = FALSE)
qqline(sample, col = "red", lwd = 2)

Interpretation

Graph shows not normal distribution

c. Compute the mean of the sampling distribution and compare it to the “population mean”

Samples_Mean<-mean(sample_means)
Samples_Mean
## [1] 6425.9
population_mean<-mean(hincp$hincp)
population_mean
## [1] 110719.6

We can clearly observe that the sample mean is much less than the population mean.

Question 5

Draw 10,000 random samples of size n = 1000 from the population and:

a. Create a histogram of the sampling distribution (hint: first you need to calculate the sample mean in each sample);

#make this example reproducible
set.seed(0)

sample<-sample(1:10000,1000,replace=T)
sample_means<-mean(sample)
head(sample_means)
## [1] 5084.031
#view first six sample means
hist(sample, main = "", xlab = "Sample Means", col = "steelblue")

Interpretation

Graph shows normal distribution

b. Create a Q-Q plot of the sampling distribution to check its normality;

qqnorm(sample, pch = 1, frame = FALSE)
qqline(sample_means, col = "red", lwd = 2)

Interpretation

Graph shows normal distribution

c. Compute the mean of the sampling distribution and compare it to the “population mean”.

Samples_Mean<-mean(sample_means)
Samples_Mean
## [1] 5084.031
population_mean<-mean(hincp$hincp)
population_mean
## [1] 110719.6

Comparison

We can clearly observe that the sample mean is much less than the population mean.

Question # 6

Confirm (or reject) the two properties CLT by comparing the histograms, Q-Q plots, and means created above, for example:

Sampling distribution for population(Household Incom )

histogram<-hist(hincp$hincp,main="Household Incom Distribution",xlab="Household Incom $")

qqnorm(hincp$hincp, pch = 1, frame = FALSE)
qqline(hincp$hincp, col = "red", lwd = 2)

The graph for population given in question 6 we can confirm for the two properties CLT by comparing that the histograms are the same or we cannot reject it.

Sample distribution for N=2

The histogram and QQ plot of sample distribution for sample size 2 is not same for the sampling distribution for population size 2 so we reject it and conclude that both are not the same.

Sample distribution for N=10

The histogram and QQ plot of sample distribution for sample size 10 is not same for the sampling distribution for population size 10 so we reject it and conclude that both are not the same.

Sample distribution for N=1000

The histogram and QQ plot of the sample distribution for sample size 1000 is the same for the sampling distribution for population size 1000 so we confirm it and conclude that both are the same.