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STA304H1S/1003HS systematic sampling

创建日期：2024-04-07 16:39:14

所属分类：统计学statistics

标签： STA304H1S

systematic sampling

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STA304H1S/1003HS

We should learn...
What is systematic sampling? It’s advantages and disadvantages...
How does systematic sampling compare with SRS?
What is repeated systematic sampling?
Shivon Sue-Chee Systematic Sampling 1
Systematic Sampling (Ch.7)
choose a random starting point and choose each 10th, or 15th, or
20th observation
Eg, every 10th person leaving a shopping centre
Eg, every 15th item in an ordered list of accounts
Eg, every half hour/fifteen minutes/ ... take an item from an
assembly line for inspection
Eg, our class heights (for n=10)
Shivon Sue-Chee Systematic Sampling 2
Class example
example: our heights (for n=10)
> N=length(height); N
[1] 217
> n=10
> #Systematic random sampling
> k=floor(N/n);k
[1] 21
> #random starting point
> set.seed(1234)
> start=sample(1:k,1); start
[1] 19
> sys<-seq(start,N,k);sys
[1] 19 40 61 82 103 124 145 166 187 208
> sample_sys<-height[sys];
> sample_sys
[1] 170.0 160.0 170.5 180.0 181.0 170.0 175.0 175.0 172.0 180.0
Shivon Sue-Chee Systematic Sampling 3
Advantages and disadvantages of systematic
sampling
easier, especially with no sampling frame
easier to organize, especially with untrained interviewers
more precise if the y ’s in the population are ordered
equivalent to a SRS if the y ’s in the population are in random order
hard to estimate the variance of y¯
so we usually use the SRS variance estimate
very biased if the y ’s in the population are cyclical, and the sampling
interval coincides with the cycle
see Figures 7.1 – 7.3
Shivon Sue-Chee Systematic Sampling 4
Population types
Shivon Sue-Chee Systematic Sampling 5
What are Random, Ordered and Periodic
populations? Examples...
Random population (ρ ≈ 0)
The elements of the population are in random order.
Ordered population (ρ < 0)
The elements of the population have values that trend upward or
downward when listed.
Periodic population (ρ ≈ 1)
The elements of the population tend to cycle upward and downward in a
regular pattern when listed.
Shivon Sue-Chee Systematic Sampling 6
Inference from systematic samples: §7.3, 7.4
comparable to SRS
When systematic sampling is nearly equivalent to SRS, we can
estimate V (y¯sy ) by V̂ (y¯).
see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)
HW: there’s a (small) mistake in formula (7.3) in the 7th edition:
what is it?
usual technique for estimating sample size n (§7.5)
Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).
Shivon Sue-Chee Systematic Sampling 7
Inference from systematic samples: §7.3, 7.4
comparable to SRS
When systematic sampling is nearly equivalent to SRS, we can
estimate V (y¯sy ) by V̂ (y¯).
see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)
HW: there’s a (small) mistake in formula (7.3) in the 7th edition:
what is it?
usual technique for estimating sample size n (§7.5)
Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).
Shivon Sue-Chee Systematic Sampling 8
Inference from systematic samples: §7.3, 7.4
comparable to SRS
When systematic sampling is nearly equivalent to SRS, we can
estimate V (y¯sy ) by V̂ (y¯).
see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)
HW: there’s a (small) mistake in formula (7.3) in the 7th edition:
what is it?
usual technique for estimating sample size n (§7.5)
Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).
Shivon Sue-Chee Systematic Sampling 9
Inference from systematic samples: §7.3, 7.4
comparable to SRS
When systematic sampling is nearly equivalent to SRS, we can
estimate V (y¯sy ) by V̂ (y¯).
see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)
HW: there’s a (small) mistake in formula (7.3) in the 7th edition:
what is it?
usual technique for estimating sample size n (§7.5)
Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).
Shivon Sue-Chee Systematic Sampling 10
Inference from systematic samples: §7.3, 7.4
comparable to SRS
When systematic sampling is nearly equivalent to SRS, we can
estimate V (y¯sy ) by V̂ (y¯).
see formulas (7.1) and (7.2); (7.5) and (7.6); (7.7) and (7.8)
HW: there’s a (small) mistake in formula (7.3) in the 7th edition:
what is it?
usual technique for estimating sample size n (§7.5)
Otherwise, SRS variance provides useful LB or UB for V (y¯sy ).
Shivon Sue-Chee Systematic Sampling 11
Inference from systematic samples: §7.3, 7.4
When are the SRS formulas okay?
if the sample is ‘similar’ to a simple random sample
i.e. population is unordered, with respect to the variable of interest
SRS variance will be an over-estimate if the population is ordered
SRS estimate will just be wrong if the population is periodic and the
sampling interval coincides with the cycle
e.g. data on store sales taken every 7th day; data on rainfall taken
every 12 months; ...
Shivon Sue-Chee Systematic Sampling 12
Inference from systematic samples: §7.3, 7.4
When are the SRS formulas okay?
if the sample is ‘similar’ to a simple random sample
i.e. population is unordered, with respect to the variable of interest
SRS variance will be an over-estimate if the population is ordered
SRS estimate will just be wrong if the population is periodic and the
sampling interval coincides with the cycle
e.g. data on store sales taken every 7th day; data on rainfall taken
every 12 months; ...
Shivon Sue-Chee Systematic Sampling 13
Inference from systematic samples: §7.3, 7.4
When are the SRS formulas okay?
if the sample is ‘similar’ to a simple random sample
i.e. population is unordered, with respect to the variable of interest
SRS variance will be an over-estimate if the population is ordered
SRS estimate will just be wrong if the population is periodic and the
sampling interval coincides with the cycle
e.g. data on store sales taken every 7th day; data on rainfall taken
every 12 months; ...
Shivon Sue-Chee Systematic Sampling 14
Why is SRS variance estimate too big if the
population is ordered?

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