HW2

1.

From Lohr Exercise 2.1. Let N = 6 and n = 3. For purposes of studying sampling distributions, assume that all population value are known.

y1 = 98, y2 = 102, y3 = 154, y4 = 133, y5 = 190, y6 = 175 We are interested in learning the population mean, so two sampling schemes are proposed.

a. (4 points)

For each of the possible samples listed below, compute the point estimate from the sample. Note plan one has 8 possible samples and plan 2 has 3 possible samples.

Plan 1. Eight possible samples may be chosen.

  1. Sample = {1,3,5}; Probability of selection = 1/8
  2. Sample = {1,3,6}; Probability of selection = 1/8
  3. Sample = {1,4,5}; Probability of selection = 1/8
  4. Sample = {1,4,6}; Probability of selection = 1/8
  5. Sample = {2,3,5}; Probability of selection = 1/8
  6. Sample = {2,3,6}; Probability of selection = 1/8
  7. Sample = {2,4,5}; Probability of selection = 1/8
  8. Sample = {2,4,6}; Probability of selection = 1/8

Plan 2. Three possible samples may be chosen

  1. Sample = {1,4,6}; Probability of selection = 1/4
  2. Sample = {1,3,6}; Probability of selection = 1/2
  3. Sample = {1,4,5}; Probability of selection = 1/4

b. (3 points)

What are the inclusion probabilities for each of the six units under Plan 1?

c. (3 points)

What are the inclusion probabilities for each of the six units under Plan 2?

d. (2 points)

What is the value of the population mean

e. (4 points)

For the estimator in plan 1, compute the expectation, variance, bias, and mean-squared error.

f. (4 points)

For the estimator in plan 2, compute the expectation, variance, bias, and mean-squared error.

g. (2 points)

Which sampling scheme do you think is better? Justify your answer.

2. (4 points)

Assume you have a one hundred households and choose to implement a systematic sampling scheme. Please write R pseudocode to carry out this procedure.