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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