Stratified sampling

Question 1

What is stratified random sampling?

Answer 1

a) a form of random sampling which involves splitting the population into distinct, non-overlapping sub-groups called strata
b) strata are homogenous

Question 2

How do you perform stratified sampling?

Answer 2

a) identify potentially relevant strata
b) take independent random samples from each stratum
c) combine results to estimated T, p, or X

Question 3

What are the advantages of stratification?

Answer 3

a) potential to reduce variance of estimates
b) more representative as each stratum is represented
c) costs may be lower
d) estimates for separate strata can be compared

Question 4

How do you calculate 𝑠𝑖 sqrd? (sample variance of i-th stratum)

Answer 4

Question 5

What does the sum of the Wi’s add to in stratified sampling?

Answer 5

1

Question 6

Why do we use weighted average in stratified sampling estimates?

Answer 6

a) sample means evaluated from larger strata have more importance (weight) and vice versa

Question 7

How do you calculate the variance of the total in stratified sampling?

Answer 7

Question 8

What do we assume when estimating in stratified sampling?

Answer 8

that the estimators are normally distributed

Question 9

How would you calculate confidence intervals with stratified sampling?

Answer 9

a) z- or t-statistics for estimating margins of error/confidence intervals for a mean or total.
b) non-normal = a z-statistic can only be used given that each 𝑛𝑖 > 30.
c) t-statistic degrees of freedom are (𝑛−𝑘), i.e. the (overall sample size – number of strata).
d) use Z-statistic for estimating margin of error/confidence interval for proportion.

Question 10

What is sample allocation in stratified?

Answer 10

splitting the overall sample size between the different strata to decide the sample size 𝑛𝑖 for each stratum.

Question 11

What are the methods of sample allocation?

Answer 11

a) proportional
b) neyman
c) optimal

Question 12

What is proportional allocation?

Answer 12

a) simplest method
b) involves sampling each stratum in proportion to its size or weight respective to the population
c) sample more from larger strata, less from smaller

Question 13

what are the requirements of proportional allocation?

Answer 13

a) means that sampling fractions are equal for each stratum (up to rounding errors)
b) ensures a representative sample;
c) straightforward and commonly used;
d) requires the stratum sizes 𝑁𝑖 or at least the stratum weights 𝑊𝑖 to be known.

Question 14

When allocating sample size, how do you round the estimate?

Answer 14

the rounding of the overall sample size and sample sizes within strata should be to the nearest whole number

Question 15

What is Neyman allocation?

Answer 15

a) chooses the 𝑛𝑖 to directly minimise the variance of the estimator 𝑥̅𝑆𝑇
b) sample more from strata with higher variability and/or larger strata.

Question 16

what are the requirements of Neyman allocation?

Answer 16

a) Requires more information than proportional allocation
b) As well as the 𝑁𝑖 (or the 𝑊𝑖), it requires 𝑆𝑖 to be known or estimated
c) For equal 𝑆𝑖, the allocation reduces to proportional allocation, so the Neyman method is more general;
d) Where the 𝑆𝑖 differ, this method is more precise than proportional allocation.

Question 17

What is relative efficiency?

Answer 17

a) measurement of the improvement in the performance of one unbiased estimator over another
b) variance of estimator a over variance of estimator b

Question 18

How do you interpret relative efficiency?

Answer 18

a) 𝑅𝐸(𝑥̅𝑆𝑇/𝑥̅𝑆𝑅𝑆 ) = 2
b) then stratified sampling was twice as efficient as SRS.

Question 19

What is the design effect?

Answer 19

a) the reciprocal of the relative efficiency
b) another way to assess which estimator, and sampling scheme, is more efficient to use.
c) variance of parameter 1 over variance of parameter 2

Question 20

How do you interpret design effect?

Answer 20

a) the design effect of stratified random sampling to SRS, when estimating a population mean would be:
𝑑𝑒𝑓𝑓(𝑥̅𝑆𝑇/𝑥̅𝑆𝑅𝑆 ) = (𝑉𝑎𝑟(𝑥̅𝑆𝑇)) / (𝑉𝑎𝑟(𝑥̅𝑆𝑅𝑆))
b) Design effects less than 1 indicate an efficient design for the sampling scheme on numerator

Question 21

What is optimal allocation?

Answer 21

a) chooses the sample sizes 𝑛𝑖 to minimise the variance
b) takes into account situations where there is a budget for conducting a survey
c) sampling costs will be different in each strata
d) sample from strata which are larger and/or have greater variability 𝑆𝑖 and/or have lower sampling costs.

Question 22

How do you calculate total cost of sampling C?

Answer 22

where ci is the cost of sampling a single population element from the i-th stratum

Question 23

what is the optimal allocation sample size formula?

Answer 23

a) where 𝛾𝑖 are the fractions of the sample size 𝑛 to be taken from the 𝑖-th stratum and ∑𝛾𝑖 =1.

Question 24

What are the optimal allocation results under certain circumstances?

Answer 24

a) For equal costs 𝑐𝑖, this method gives the same as Neyman allocation.
b) For equal costs and equal 𝑆𝑖, this method gives the same as proportional allocation
c) so optimal allocation is the most general of the 3 methods of allocation

Question 25

Whats the difference between the different types of allocation?

Answer 25

a) Proportional: Use when 𝑆𝑖 are equal and 𝑐𝑖 are equal across the strata, when stratified random sampling is considered. b) Neyman: Improves on proportional allocation when the 𝑆𝑖 differ (with equal sampling costs across strata), when stratified random sampling is considered. c) Optimal: Should be used when known sampling costs differ between strata (regardless if 𝑆𝑖 are equal or different).