Systematic Sampling

Question 1

Which population is simple random sampling best suited to?

Answer 1

homogenous not heterogenous which many populations are

Question 1

Why is systematic sampling often preferred over simple random?

Answer 1

a) SRS can be time-consuming or costly
b) systematic sampling is simple, quick, and convenient

Question 2

How do you perform systematic sampling?

Answer 2

1) split population into n intervals
2) randomly select the first item in first interval
3) select every k-th item after
4) this gives a 1 in k systematic sample where k is sampling interval
5) k = N/n

Question 3

How do you use systematic sampling when N is not an exact multiple of n?

Answer 3

a) 𝑁=(𝑛−1)𝑘+𝑡, 0 < 𝑡 < 𝑘
b) we sample the first element from the first t items in the population then every kth one thereafter
c) this ensures that the nth sample item is selected before the last item in the population is reached.

Question 4

How do you use systematic sampling when N is unknown?

Answer 4

a) guess an appropriate sampling interval to evenly spread items within sampling frame

Question 5

What is the inclusion probability in systematic sampling?

Answer 5

a) all sample sizes of n are not equally likely to be selected; different sets may have unequal inclusion probabilities
b) therefore, we cannot estimate the variance of estimators from a single sample

Question 6

How do we calculate variance of estimators in systematic sampling?

Answer 6

a) approximate the variance
b) or take repeated samples

Question 7

what value does 𝜌 take when estimating systematic sample mean variance if population is listed in a random order concerning variable X?

Answer 7

𝜌 ~ = 0

> so var(x sys) = var(x srs)

Question 8

what value does 𝜌 take when estimating systematic sample mean variance if population is listed in order concerning variable X?

Answer 8

a) items within a sample will tend to be dissimilar
b) 𝜌 < 0
c) var(x sys) < var(x srs) = systematic is more precise

Question 9

what value does 𝜌 take when estimating systematic sample mean variance if population is ordered periodically (repeating pattern) concerning variable X?

Answer 9

a) items in a sample will be similar
b) 𝜌 > 0
c) var(x srs) < var(x sys) = SRS produces a more precise estimate

Question 10

How do you estimate population total in systematic sampling?

Answer 10

a) follows the same convention as SRS
b) T sys = N x (sample mean sys)

Question 11

How do you estimate population proportion in systematic sampling?

Answer 11

a) follows SRS method
b) assuming 𝜌 ~ = 0 (randomly ordered population) for variance

Question 12

How do you choose sample size via systematic sampling?

Answer 12

a) using SRS formulae
b) results should be approximately the same if population is randomly ordered
c) n may be too small for periodically ordered
d) n may be too large for increasing/decreasing ordered population

Question 13

How do you perform repeated systematic sampling?

Answer 13

1) Pick how many repeated samples you want — call this number 𝑛𝑠
2) Split your total sample size 𝑛 into 𝑛𝑠 smaller samples, each of size 𝑛 / 𝑛𝑠
3) Use a bigger step size (called 𝑘′) between selections, so you’re not picking items too close together.
4) This new step size is: 𝑘′ = original𝑘 × 𝑛𝑠
5) Choose a random start point for each of the 𝑛𝑠 smaller samples.
6) Take every 𝑘′th item after each start point

Question 14

What is different with the variance formula for repeated systematic sampling?

Answer 14

𝑛𝑠 replaces 𝑛