Finite Population Sampling

Question 1

Give two reasons why people choose to sample.

Answer 1

1. Cheaper and faster than full census (could be impossible)
2. May be more accurate (when census’s get too large quality control is harder)

Question 2

What does N stand for?

Answer 2

Number of inidividuals in the population

Question 3

What does x_i stand for?

Answer 3

The value of x for individul i

Question 4

What does θ stand for?

Answer 4

Population parameter, function of the population values.

Question 5

Name four population parameters.

Answer 5

1. Population average
2. Population total
3. Population variance
4. Population standard deviation

Question 6

What is the equation for population average?

Answer 6

Question 7

What is the equation for population total?

Answer 7

Question 8

What is the equation for population variance?

Answer 8

Question 9

What is the difference between N and n?

Answer 9

• N - Population size
• n - Sample size

Question 10

What is the number of possible samples, when using a simple random sample without replacement?

Answer 10

Question 11

Define a statistic.

Answer 11

A statistic is any function of the sample observations.

Question 12

Define a sampling distribution.

Answer 12

The sampling distribution of a statistic is:

1. the set of possible values that the statistic can take
2. the probability that the statistic takes each values

Question 13

Are samples with smaller or larger variacnes more reliable?

Answer 13

Smaller

Question 14

What is the difference between Ȳ_n and 𝜇?

Answer 14

• Ȳ_n is the sample average
• 𝜇 is the population average

Question 15

Prove 𝔼(Ȳ_n) = 𝜇.

Answer 15

Question 16

What does 𝔼(Ȳn) = 𝜇 show?

Answer 16

The sample mean is an unbiased estimator for the population mean.

Question 17

Define bias and therfore unbiased.

Answer 17

If statistic S estimates population parameter θ, then the bias of S is

b(S, θ) = 𝔼(S) - θ

If the bias is zero, then S is called unbiased.

Question 18

What does σ_Ȳn stand for?

Answer 18

Standard error of estimator Ȳ_n.

Question 19

Prove

Answer 19

Question 20

When N >> n what can you say about this inequality?

Answer 20

Question 21

For large N what does var(Ȳ_n) approximately equal?

Answer 21

Question 22

What does f stand for in this equation?

Answer 22

f is the sampling fraciton, n/N

Question 23

What is the equation for sample variance?

Answer 23

Question 24

What does s_Ȳ stand for?

Answer 24

Estimated standard error of Ȳ

Question 25

Complete this therom.

1. s² is almost unbiased for σ^2,, as …
2. s_Ȳ² is an unbiased estimator for σ_Ȳ², where …

Answer 25

Question 26

What are the possible values of x_i when talking about binary outcomes?

Answer 26

0 or 1

Question 27

Bianry outcomes - If the proportion of indviduals in the population with x = 1 is p what is 𝜇 and σ²?

Answer 27

• 𝜇 = p
• σ² = p(1-p)

Question 28

If p^{^} is the proportion of individuals in sample with x=1 then what does Ȳ equal?

Answer 28

Question 29

If p^ is the proportion of individuals in sample with x=1 then what does s² equal?

Answer 29

Question 30

If p^ is the proportion of individuals in sample with x=1 then what does σp^2 equal?

Answer 30

Question 31

What is σ_p^² smaller than?

Answer 31

Question 32

What is the equation for the unbiased estimator for σ_p^²?

Answer 32