Sampling Distributions--Finite Populations

The sampling distribution of a given statistic is its probability distribution when we consider all possible random samples of size n from a population. The shape, center, and spread of the distribution depend on…

• the statistic being modeled
• the population distribution
• the particular sample size
• the type of sampling used

The standard deviation of a sampling distribution is its standard error (sometimes abbreviated SE).

A one-time discount card for a store you like has four possible values that are equally likely: 0%, 5%, 10%, or 25%. You receive two randomly chosen cards that are guaranteed to be different. What is the larger (max) value you get? All samples of size n = 2 without replacement are shown below.

require(combinat)
cards   <- c(0,5,10,25)
samples <- combn(cards, 2)
prmatrix(samples, rowlab=rep("",2))

 [,1] [,2] [,3] [,4] [,5] [,6]
    0    0    0    5    5   10
    5   10   25   10   25   25

We need to find the maximum of all six possible samples.

s.max <- numeric(6)
for(i in 1:6)
{
 s.max[i]<- max(samples[,i])
}
print(s.max)

[1]  5 10 25 10 25 25

What are the possibilities and probabilities?

pmf <- prop.table(table(s.max))
print(round(pmf,4))

s.max
     5     10     25 
0.1667 0.3333 0.5000 

The max discount of your two cards is most likely to be 25%.

What are E(X), Var(X), and SE(X) for the sampling distribution?

EX <- mean(s.max)
VX <- sum((s.max-EX)^2)/length(s.max)
SX <- sqrt(VX)
PARAMS <- c(EX,VX,SX)
names(PARAMS) <- c("E(X)","Var(X)","SE(X)")
print(round(PARAMS,4))

   E(X)  Var(X)   SE(X) 
16.6667 72.2222  8.4984