Paper_Clips

Imagine a population of paper clips of varying size. Are the following true?

\(\frac{ \bar{X} - \mu }{\frac{\sigma}{\sqrt{n}}} \sim \mathcal{Z}\)

\(\frac{ \bar{X} - \mu }{\frac{s}{\sqrt{n}}} \sim \mathcal{Z}\) ?????

population <-rnorm(100000,5,2)
len <- 1000000
n <- 5

x <- vector(mode="numeric", length = len)
i <- 1

for( i in 1:len) {
        
        x[i] <-( mean(sample(population,n)) - 5 ) / (2 / sqrt(n))
        
}

hist(x)

How can we tell if this is the Z distribution?

quantile(x,.975)

##  97.5% 
## 1.9551

Now the realistic situation when we do not know the population standard deviation.

#population <-rnorm(10000,5,2)
len <- 1000000
#n <- 5

y <- vector(mode="numeric", length = len)
j <- 1

for( j in 1:len) {
        data <- sample(population,n)
       
        y[j] <-( mean(data) - 5 ) / (sd(data) / sqrt(n))
       
}

hist(y)

How can we tell if this is the Z distribution?

quantile(y,.975)

##    97.5% 
## 2.757499