Sampling

Probability Sampling Nonprobability Sampling

Probability Sampling

• Sampling element• Population• Target population• Sampling frame• Sampling ratio

There is a classic Jimmy Stewart movie, Magic Town, about "Grandview," a small town in the Midwest that is a perfect statistical microcosm of the United States, a place where the citizens' opinions match perfectly with Gallup polls of the entire nation. A pollster (Jimmy Stewart), secretly uses surveys from this "mathematical miracle" as a shortcut to predicting public opinion. Instead of collecting a national sample, he can more quickly and cheaply collect surveys from this single small town. The character played by Jane Wyman, a newspaper editor, finds out what is going on and publishes her discovery. As a result the national media descend upon the town, which becomes, overnight, "the public opinion capital of the U.S."

Probability Sampling

Sampling Distribution

Probability Sampling

• Random sample• Sampling error

• Four Ways to Sample Randomly– Simple Random– Systematic– Stratified Sampling– Cluster Sampling

Random Sample

• Sampling Error:

𝑥=0.5

𝜇=0.5625

𝑥=0.75

Variation Component

Sample size Component

R Sessiondata=c(1,1,0,0,0,0,1,1,0,1,1,0,1,1,1,0)population.mean=mean(data)

#samples of size 5a.sample=sample(x=data,size=5,replace=FALSE)a.mean=mean(a.sample)#another sampleb.sample=sample(data,5,FALSE)b.mean=mean(b.sample)

#Distribution of sample mean#We need to sample lots of timessim.runs=100mean.sample=NAfor (i in 1:sim.runs){ sample.data=sample(data,5,FALSE) mean.sample[i]=mean(sample.data)}hist(mean.sample,breaks=4)

Histogram of mean.sample

mean.sample

Frequency

0.0 0.2 0.4 0.6 0.8 1.0

010

2030

Sampling Distribution and Sampling Error

Let’s first see what mathematics has to say.

1. According to Law of Large Numbers:

As sample size increases (approaches to ) sample mean approaches to population mean, in mathematical symbols

2. According to Central Limit Theorem

As the number of samples (not the sample size, this time) increases then sample mean has a normal distribution with mean and standard deviation . Mathematically we say,

Sampling and Confidence

x

𝜇 𝑥𝑢𝑥 𝑙 𝑥

𝑥−(𝑧∗ 𝜎√𝑛 )≤𝜇≤𝑥+(𝑧∗ 𝜎

√𝑛 )1. Confidence information is in z.2. can be replaced by .

Important Concepts in Sampling

𝑥−𝜇≤(𝑧∗ 𝜎√𝑛 )

The value of z depends on confidence

Margin of Error

Finite Population Correction Factor

Sam

plin

g er

ror

Next: Sample size

Other Probability Sampling Designs