Sampling Distribution of Sample Mean

Example 1 (Data Set 21)

1. Upload Data Set 21 to R in previously saved .csv format.
2. Use Data Set 21 to construct a histogram of DEPTHS of 600 earthquakes.
3. Select 10000 random samples of size=50 from the DEPTHS variable, calculate the mean of each and construct a histogram of the sampling distribution of the sample means.
   SOLUTION

EarthQ <- read.csv("~/Desktop/csv/21 - Earthquakes.csv")

attach(EarthQ)

head(EarthQ)

##   MAGNITUDE DEPTH
## 1      2.45   0.7
## 2      3.62   6.0
## 3      3.06   7.0
## 4      3.30   5.4
## 5      1.09   0.5
## 6      3.10   0.0

2.  

breaks <-seq(0,45,by=5)
hist(DEPTH,breaks, col="red",xlab = "Depth [km]",ylab = "Frequency",main = "Histogram of Earthquakes Depths")

mean(DEPTH)

## [1] 5.822

sd(DEPTH)

## [1] 4.927049

fifty.depths <- function() {
    depth.S <- sample(DEPTH,
    size = 50,replace = TRUE)
    return(mean(depth.S))
}

 sim1 <-replicate(n=10000,expr=fifty.depths())

head(sim1)

## [1] 5.540 6.576 5.224 6.800 5.262 4.952

summary(sim1)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   3.650   5.338   5.798   5.825   6.270   8.742

sd(sim1)

## [1] 0.695623

breaks <-seq(2.5,10.5,by=0.5)
hist(sim1,breaks,xlab ="Mean[km]",ylab="Frequency",col="red",border = "green",main="Sampling Distribution of the Sample Mean")

Things to ponder:

1. the shape of DEPTH histogram
2. values of the mean(DEPTH) and sd(DEPTH)
3. shape of means(DEPTH) sampling distribution
4. values of the mean of the sampling distribution and standard deviation of the sampling distribution